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Technical: Taproot: Why Activate?

This is a follow-up on https://old.reddit.com/Bitcoin/comments/hqzp14/technical_the_path_to_taproot_activation/
Taproot! Everybody wants it!! But... you might ask yourself: sure, everybody else wants it, but why would I, sovereign Bitcoin HODLer, want it? Surely I can be better than everybody else because I swapped XXX fiat for Bitcoin unlike all those nocoiners?
And it is important for you to know the reasons why you, o sovereign Bitcoiner, would want Taproot activated. After all, your nodes (or the nodes your wallets use, which if you are SPV, you hopefully can pester to your wallet vendoimplementor about) need to be upgraded in order for Taproot activation to actually succeed instead of becoming a hot sticky mess.
First, let's consider some principles of Bitcoin.
I'm sure most of us here would agree that the above are very important principles of Bitcoin and that these are principles we would not be willing to remove. If anything, we would want those principles strengthened (especially the last one, financial privacy, which current Bitcoin is only sporadically strong with: you can get privacy, it just requires effort to do so).
So, how does Taproot affect those principles?

Taproot and Your /Coins

Most HODLers probably HODL their coins in singlesig addresses. Sadly, switching to Taproot would do very little for you (it gives a mild discount at spend time, at the cost of a mild increase in fee at receive time (paid by whoever sends to you, so if it's a self-send from a P2PKH or bech32 address, you pay for this); mostly a wash).
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash, so the Taproot output spends 12 bytes more; spending from a P2WPKH requires revealing a 32-byte public key later, which is not needed with Taproot, and Taproot signatures are about 9 bytes smaller than P2WPKH signatures, but the 32 bytes plus 9 bytes is divided by 4 because of the witness discount, so it saves about 11 bytes; mostly a wash, it increases blockweight by about 1 virtual byte, 4 weight for each Taproot-output-input, compared to P2WPKH-output-input).
However, as your HODLings grow in value, you might start wondering if multisignature k-of-n setups might be better for the security of your savings. And it is in multisignature that Taproot starts to give benefits!
Taproot switches to using Schnorr signing scheme. Schnorr makes key aggregation -- constructing a single public key from multiple public keys -- almost as trivial as adding numbers together. "Almost" because it involves some fairly advanced math instead of simple boring number adding, but hey when was the last time you added up your grocery list prices by hand huh?
With current P2SH and P2WSH multisignature schemes, if you have a 2-of-3 setup, then to spend, you need to provide two different signatures from two different public keys. With Taproot, you can create, using special moon math, a single public key that represents your 2-of-3 setup. Then you just put two of your devices together, have them communicate to each other (this can be done airgapped, in theory, by sending QR codes: the software to do this is not even being built yet, but that's because Taproot hasn't activated yet!), and they will make a single signature to authorize any spend from your 2-of-3 address. That's 73 witness bytes -- 18.25 virtual bytes -- of signatures you save!
And if you decide that your current setup with 1-of-1 P2PKH / P2WPKH addresses is just fine as-is: well, that's the whole point of a softfork: backwards-compatibility; you can receive from Taproot users just fine, and once your wallet is updated for Taproot-sending support, you can send to Taproot users just fine as well!
(P2WPKH and P2WSH -- SegWit v0 -- addresses start with bc1q; Taproot -- SegWit v1 --- addresses start with bc1p, in case you wanted to know the difference; in bech32 q is 0, p is 1)
Now how about HODLers who keep all, or some, of their coins on custodial services? Well, any custodial service worth its salt would be doing at least 2-of-3, or probably something even bigger, like 11-of-15. So your custodial service, if it switched to using Taproot internally, could save a lot more (imagine an 11-of-15 getting reduced from 11 signatures to just 1!), which --- we can only hope! --- should translate to lower fees and better customer service from your custodial service!
So I think we can say, very accurately, that the Bitcoin principle --- that YOU are in control of your money --- can only be helped by Taproot (if you are doing multisignature), and, because P2PKH and P2WPKH remain validly-usable addresses in a Taproot future, will not be harmed by Taproot. Its benefit to this principle might be small (it mostly only benefits multisignature users) but since it has no drawbacks with this (i.e. singlesig users can continue to use P2WPKH and P2PKH still) this is still a nice, tidy win!
(even singlesig users get a minor benefit, in that multisig users will now reduce their blockchain space footprint, so that fees can be kept low for everybody; so for example even if you have your single set of private keys engraved on titanium plates sealed in an airtight box stored in a safe buried in a desert protected by angry nomads riding giant sandworms because you're the frickin' Kwisatz Haderach, you still gain some benefit from Taproot)
And here's the important part: if P2PKH/P2WPKH is working perfectly fine with you and you decide to never use Taproot yourself, Taproot will not affect you detrimentally. First do no harm!

Taproot and Your Contracts

No one is an island, no one lives alone. Give and you shall receive. You know: by trading with other people, you can gain expertise in some obscure little necessity of the world (and greatly increase your productivity in that little field), and then trade the products of your expertise for necessities other people have created, all of you thereby gaining gains from trade.
So, contracts, which are basically enforceable agreements that facilitate trading with people who you do not personally know and therefore might not trust.
Let's start with a simple example. You want to buy some gewgaws from somebody. But you don't know them personally. The seller wants the money, you want their gewgaws, but because of the lack of trust (you don't know them!! what if they're scammers??) neither of you can benefit from gains from trade.
However, suppose both of you know of some entity that both of you trust. That entity can act as a trusted escrow. The entity provides you security: this enables the trade, allowing both of you to get gains from trade.
In Bitcoin-land, this can be implemented as a 2-of-3 multisignature. The three signatories in the multisgnature would be you, the gewgaw seller, and the escrow. You put the payment for the gewgaws into this 2-of-3 multisignature address.
Now, suppose it turns out neither of you are scammers (whaaaat!). You receive the gewgaws just fine and you're willing to pay up for them. Then you and the gewgaw seller just sign a transaction --- you and the gewgaw seller are 2, sufficient to trigger the 2-of-3 --- that spends from the 2-of-3 address to a singlesig the gewgaw seller wants (or whatever address the gewgaw seller wants).
But suppose some problem arises. The seller gave you gawgews instead of gewgaws. Or you decided to keep the gewgaws but not sign the transaction to release the funds to the seller. In either case, the escrow is notified, and if it can sign with you to refund the funds back to you (if the seller was a scammer) or it can sign with the seller to forward the funds to the seller (if you were a scammer).
Taproot helps with this: like mentioned above, it allows multisignature setups to produce only one signature, reducing blockchain space usage, and thus making contracts --- which require multiple people, by definition, you don't make contracts with yourself --- is made cheaper (which we hope enables more of these setups to happen for more gains from trade for everyone, also, moon and lambos).
(technology-wise, it's easier to make an n-of-n than a k-of-n, making a k-of-n would require a complex setup involving a long ritual with many communication rounds between the n participants, but an n-of-n can be done trivially with some moon math. You can, however, make what is effectively a 2-of-3 by using a three-branch SCRIPT: either 2-of-2 of you and seller, OR 2-of-2 of you and escrow, OR 2-of-2 of escrow and seller. Fortunately, Taproot adds a facility to embed a SCRIPT inside a public key, so you can have a 2-of-2 Taprooted address (between you and seller) with a SCRIPT branch that can instead be spent with 2-of-2 (you + escrow) OR 2-of-2 (seller + escrow), which implements the three-branched SCRIPT above. If neither of you are scammers (hopefully the common case) then you both sign using your keys and never have to contact the escrow, since you are just using the escrow public key without coordinating with them (because n-of-n is trivial but k-of-n requires setup with communication rounds), so in the "best case" where both of you are honest traders, you also get a privacy boost, in that the escrow never learns you have been trading on gewgaws, I mean ewww, gawgews are much better than gewgaws and therefore I now judge you for being a gewgaw enthusiast, you filthy gewgawer).

Taproot and Your Contracts, Part 2: Cryptographic Boogaloo

Now suppose you want to buy some data instead of things. For example, maybe you have some closed-source software in trial mode installed, and want to pay the developer for the full version. You want to pay for an activation code.
This can be done, today, by using an HTLC. The developer tells you the hash of the activation code. You pay to an HTLC, paying out to the developer if it reveals the preimage (the activation code), or refunding the money back to you after a pre-agreed timeout. If the developer claims the funds, it has to reveal the preimage, which is the activation code, and you can now activate your software. If the developer does not claim the funds by the timeout, you get refunded.
And you can do that, with HTLCs, today.
Of course, HTLCs do have problems:
Fortunately, with Schnorr (which is enabled by Taproot), we can now use the Scriptless Script constuction by Andrew Poelstra. This Scriptless Script allows a new construction, the PTLC or Pointlocked Timelocked Contract. Instead of hashes and preimages, just replace "hash" with "point" and "preimage" with "scalar".
Or as you might know them: "point" is really "public key" and "scalar" is really a "private key". What a PTLC does is that, given a particular public key, the pointlocked branch can be spent only if the spender reveals the private key of the given public key to you.
Another nice thing with PTLCs is that they are deniable. What appears onchain is just a single 2-of-2 signature between you and the developemanufacturer. It's like a magic trick. This signature has no special watermarks, it's a perfectly normal signature (the pledge). However, from this signature, plus some datta given to you by the developemanufacturer (known as the adaptor signature) you can derive the private key of a particular public key you both agree on (the turn). Anyone scraping the blockchain will just see signatures that look just like every other signature, and as long as nobody manages to hack you and get a copy of the adaptor signature or the private key, they cannot get the private key behind the public key (point) that the pointlocked branch needs (the prestige).
(Just to be clear, the public key you are getting the private key from, is distinct from the public key that the developemanufacturer will use for its funds. The activation key is different from the developer's onchain Bitcoin key, and it is the activation key whose private key you will be learning, not the developer's/manufacturer's onchain Bitcoin key).
So:
Taproot lets PTLCs exist onchain because they enable Schnorr, which is a requirement of PTLCs / Scriptless Script.
(technology-wise, take note that Scriptless Script works only for the "pointlocked" branch of the contract; you need normal Script, or a pre-signed nLockTimed transaction, for the "timelocked" branch. Since Taproot can embed a script, you can have the Taproot pubkey be a 2-of-2 to implement the Scriptless Script "pointlocked" branch, then have a hidden script that lets you recover the funds with an OP_CHECKLOCKTIMEVERIFY after the timeout if the seller does not claim the funds.)

Quantum Quibbles!

Now if you were really paying attention, you might have noticed this parenthetical:
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash...)
So wait, Taproot uses raw 32-byte public keys, and not public key hashes? Isn't that more quantum-vulnerable??
Well, in theory yes. In practice, they probably are not.
It's not that hashes can be broken by quantum computes --- they're still not. Instead, you have to look at how you spend from a P2WPKH/P2PKH pay-to-public-key-hash.
When you spend from a P2PKH / P2WPKH, you have to reveal the public key. Then Bitcoin hashes it and checks if this matches with the public-key-hash, and only then actually validates the signature for that public key.
So an unconfirmed transaction, floating in the mempools of nodes globally, will show, in plain sight for everyone to see, your public key.
(public keys should be public, that's why they're called public keys, LOL)
And if quantum computers are fast enough to be of concern, then they are probably fast enough that, in the several minutes to several hours from broadcast to confirmation, they have already cracked the public key that is openly broadcast with your transaction. The owner of the quantum computer can now replace your unconfirmed transaction with one that pays the funds to itself. Even if you did not opt-in RBF, miners are still incentivized to support RBF on RBF-disabled transactions.
So the extra hash is not as significant a protection against quantum computers as you might think. Instead, the extra hash-and-compare needed is just extra validation effort.
Further, if you have ever, in the past, spent from the address, then there exists already a transaction indelibly stored on the blockchain, openly displaying the public key from which quantum computers can derive the private key. So those are still vulnerable to quantum computers.
For the most part, the cryptographers behind Taproot (and Bitcoin Core) are of the opinion that quantum computers capable of cracking Bitcoin pubkeys are unlikely to appear within a decade or two.
So:
For now, the homomorphic and linear properties of elliptic curve cryptography provide a lot of benefits --- particularly the linearity property is what enables Scriptless Script and simple multisignature (i.e. multisignatures that are just 1 signature onchain). So it might be a good idea to take advantage of them now while we are still fairly safe against quantum computers. It seems likely that quantum-safe signature schemes are nonlinear (thus losing these advantages).

Summary

I Wanna Be The Taprooter!

So, do you want to help activate Taproot? Here's what you, mister sovereign Bitcoin HODLer, can do!

But I Hate Taproot!!

That's fine!

Discussions About Taproot Activation

submitted by almkglor to Bitcoin [link] [comments]

small passive income while browsing the web

Introduction
If you were on the internet in the late 1990s, you might remember companies like "AllAdvantage" that promised to pay you to browse the web. You could install a program that tracked your browsing and showed you targeted ads at the top of the screen, then "AllAdvantage" would give you a cut of the ad revenue you generated.
These schemes largely disappeared after the dot-com crash. But Brendan Eich, the creator of the JavaScript programming language and cofounder and former CTO of Mozilla, thinks his company Brave Software has found a way to revive that old idea.
What is it?
Brave makes a browser based on Google Chrome that blocks tracking scripts and other technologies that spy on your online activity. As a result, it also blocks many web ads; if you visit any website using the Brave browser, you won’t see any ads. But Brave will give users the option to see ads that Eich says will respect your privacy. The ads will appear as desktop notifications, he says, not as replacements for the ads the Brave browser blocks. So you still won’t see ads on any website, but you might see them on the right lower corner of your screen. If you choose to see these ads, you’ll get 70 percent of the revenue they generate.
Eich hopes Brave can solve two of the web's most vexing problems the privacy and revenue problem by turning the traditional digital advertising model on its head. Today, ad networks pay sites for ad space and web browsers like Brave and Chrome deliver content from those publishers to users. Brave is trying to put the browser in the center of the advertising experience. Instead of paying publishers directly, ad networks would pay Brave, which will pass part of the money to users and keep a cut for itself.
By handling advertising in the browser on your device, Brave says it will be able to target ads without sending your data to the cloud, and protect your privacy. When you interact with an ad on Brave, the browser sends notice to the company's servers, but doesn't include any identifying information. Eich sees four sets of winners: browser makers get paid; users get paid, and get more privacy; advertisers can target pitches without running afoul of European privacy regulations; and publishers can survive in a world where many users are installing ad blockers.
Publishers and ad networks might bristle at the idea of putting browser makers in the middle of their business. But in recent years browsers have taken a more active role in shaping the web, instead of merely displaying a website’s content. Chrome now blocks ads on a small number of sites with particularly egregious advertising practices, while browsers like Firefox and Safari have added privacy protections. Meanwhile, browser plugins are giving users more control over their experience. There are Chrome extensions, for example, that let you change Facebook's color scheme, or change the way images are displayed on Pinterest. And of course there are extensions that block all ads.
Trying to win advertisers and publishers to a new model isn't Brave's only challenge. It also needs users. Eich says Brave has 15 million users and is growing.
Brave will give users a 70 percent cut of its advertising revenue, which Eich estimates could work out to about $10 a month. Brave will pay users with its own bitcoin-style "cryptocurrency” called Basic Attention Tokens or BAT, which has traded for as little as 24 cents over the past 12 months, according to CoinMarketCap. You can exchange the BAT you have received for viewing ads into USD, EUR, GBP, CHF and many more currencies.
The company offers a service through the cryptocurrency exchange Uphold to allow users to change, sell and buy BAT or donate it to publishers, and for publishers to exchange the BAT they receive for dollars. Advertisers like HomeDepot or recent campaigns included brands such as Verizon, Newegg, Chipotle, and PayPal/Honey, in addition to earlier campaigns by Amazon, Harry’s Razors, Intel, CBS, KIND snacks, Logitech, Lenovo, Grubhub, Belkin, Quickbooks, Evernote and some of cryptocurrency related companies, will be able to buy ads either with BAT or with traditional currencies.
Eich says Brave opted to create its own tokens using the Ethereum cryptocurrency platform in part to avoid regulatory requirements, such as verifying users' identifies, that partners like Uphold are better equipped to handle.
Estimated revenue? (depending on the country you live in the revenue can be higher or lower)
I made around 3oo$ so far this year using 3 devices, just for viewing some ads.
5 months so far july is not included if you calculate it down for 1 device, 100$/5months = 20$ a month just for viewing ads, you would need to buy risky stocks worth of 2000$ to get the same amount per month.
can only recommend everyone to try it, not every country has the same number of advertisers so you probably get the most out of it when you live in the USA.
If you are interested here is a quick guide how to set it up to get the max amount out of Brave:
Quickstartguide:
1 Download brave here
2 Activate the reward system (gif link below)Gif link
3 go into the settings an deactivate auto contribution and activate 5 ads per hour (image link below)image link
4 Create an Account on Uphold and connect it with your BraveBrowser.
Now you are good to go and can make some money on something you do anyway.
I hope this helps some folks in the community to make some extra bucks.
edit1:you can find more infos and support here:brave_browser & BATProject or www.brave.com
edit2:the earnings are depenging on the number of devices you are using and were you are living. Best paying countries: United States (69) United Kingdom (39) Canada (36) Australia (35) New Zealand (26) Germany (21) Ireland (21) France (18)( the number next to the country are the companies that are running ads on brave for this particular country, the more companies the more revenue )
you can find a full list with all countries and campaigns here: https://brave.com/transparency/
edit3:You don't need to browse to a certain website to receive ads, just browse as you are used to, play browser games, watch videos on youtube or do whatever you want.Sometimes Ads appear on the startpage looks like that https://i.imgur.com/5tohhRc.jpg and after some time on the right lower corner a clickable pop-up appears looks like that->https://i.imgur.com/CTGdVsu.png
edit4:If you want to import your bookmarks and settings from your old browser:on the right top corner of the browser is a button ->https://i.imgur.com/oi8EAri.jpg click it > than on settings > and than you got the option to import bookmarks and settings from your old browser.
If you want to sync brave between devices and for backups:type brave://flags/ into the adressbar and than brave sync into the search bar and acticate itif its enabled it should look like this https://imgur.com/a/tCMDgDjthan just click on sync ->https://i.imgur.com/oi8EAri.jpg
here is a guide ->https://support.brave.com/hc/en-us/articles/360021218111-How-do-I-set-up-Sync
edit5: Don't keep your BAT from free token grants to long in your browser, always send your bat to an external wallet or exchange like uphold, only tokens from free token grants have an expire date if they dont get used they go back to the bat pool. you can find more infos about this here -> https://support.brave.com/hc/en-us/articles/360018305731-Why-does-my-BAT-have-an-expiration-date-
submitted by OnlyReveal6 to beermoneyglobal [link] [comments]

[ Bitcoin ] Technical: Taproot: Why Activate?

Topic originally posted in Bitcoin by almkglor [link]
This is a follow-up on https://old.reddit.com/Bitcoin/comments/hqzp14/technical_the_path_to_taproot_activation/
Taproot! Everybody wants it!! But... you might ask yourself: sure, everybody else wants it, but why would I, sovereign Bitcoin HODLer, want it? Surely I can be better than everybody else because I swapped XXX fiat for Bitcoin unlike all those nocoiners?
And it is important for you to know the reasons why you, o sovereign Bitcoiner, would want Taproot activated. After all, your nodes (or the nodes your wallets use, which if you are SPV, you hopefully can pester to your wallet vendoimplementor about) need to be upgraded in order for Taproot activation to actually succeed instead of becoming a hot sticky mess.
First, let's consider some principles of Bitcoin.
I'm sure most of us here would agree that the above are very important principles of Bitcoin and that these are principles we would not be willing to remove. If anything, we would want those principles strengthened (especially the last one, financial privacy, which current Bitcoin is only sporadically strong with: you can get privacy, it just requires effort to do so).
So, how does Taproot affect those principles?

Taproot and Your /Coins

Most HODLers probably HODL their coins in singlesig addresses. Sadly, switching to Taproot would do very little for you (it gives a mild discount at spend time, at the cost of a mild increase in fee at receive time (paid by whoever sends to you, so if it's a self-send from a P2PKH or bech32 address, you pay for this); mostly a wash).
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash, so the Taproot output spends 12 bytes more; spending from a P2WPKH requires revealing a 32-byte public key later, which is not needed with Taproot, and Taproot signatures are about 9 bytes smaller than P2WPKH signatures, but the 32 bytes plus 9 bytes is divided by 4 because of the witness discount, so it saves about 11 bytes; mostly a wash, it increases blockweight by about 1 virtual byte, 4 weight for each Taproot-output-input, compared to P2WPKH-output-input).
However, as your HODLings grow in value, you might start wondering if multisignature k-of-n setups might be better for the security of your savings. And it is in multisignature that Taproot starts to give benefits!
Taproot switches to using Schnorr signing scheme. Schnorr makes key aggregation -- constructing a single public key from multiple public keys -- almost as trivial as adding numbers together. "Almost" because it involves some fairly advanced math instead of simple boring number adding, but hey when was the last time you added up your grocery list prices by hand huh?
With current P2SH and P2WSH multisignature schemes, if you have a 2-of-3 setup, then to spend, you need to provide two different signatures from two different public keys. With Taproot, you can create, using special moon math, a single public key that represents your 2-of-3 setup. Then you just put two of your devices together, have them communicate to each other (this can be done airgapped, in theory, by sending QR codes: the software to do this is not even being built yet, but that's because Taproot hasn't activated yet!), and they will make a single signature to authorize any spend from your 2-of-3 address. That's 73 witness bytes -- 18.25 virtual bytes -- of signatures you save!
And if you decide that your current setup with 1-of-1 P2PKH / P2WPKH addresses is just fine as-is: well, that's the whole point of a softfork: backwards-compatibility; you can receive from Taproot users just fine, and once your wallet is updated for Taproot-sending support, you can send to Taproot users just fine as well!
(P2WPKH and P2WSH -- SegWit v0 -- addresses start with bc1q; Taproot -- SegWit v1 --- addresses start with bc1p, in case you wanted to know the difference; in bech32 q is 0, p is 1)
Now how about HODLers who keep all, or some, of their coins on custodial services? Well, any custodial service worth its salt would be doing at least 2-of-3, or probably something even bigger, like 11-of-15. So your custodial service, if it switched to using Taproot internally, could save a lot more (imagine an 11-of-15 getting reduced from 11 signatures to just 1!), which --- we can only hope! --- should translate to lower fees and better customer service from your custodial service!
So I think we can say, very accurately, that the Bitcoin principle --- that YOU are in control of your money --- can only be helped by Taproot (if you are doing multisignature), and, because P2PKH and P2WPKH remain validly-usable addresses in a Taproot future, will not be harmed by Taproot. Its benefit to this principle might be small (it mostly only benefits multisignature users) but since it has no drawbacks with this (i.e. singlesig users can continue to use P2WPKH and P2PKH still) this is still a nice, tidy win!
(even singlesig users get a minor benefit, in that multisig users will now reduce their blockchain space footprint, so that fees can be kept low for everybody; so for example even if you have your single set of private keys engraved on titanium plates sealed in an airtight box stored in a safe buried in a desert protected by angry nomads riding giant sandworms because you're the frickin' Kwisatz Haderach, you still gain some benefit from Taproot)
And here's the important part: if P2PKH/P2WPKH is working perfectly fine with you and you decide to never use Taproot yourself, Taproot will not affect you detrimentally. First do no harm!

Taproot and Your Contracts

No one is an island, no one lives alone. Give and you shall receive. You know: by trading with other people, you can gain expertise in some obscure little necessity of the world (and greatly increase your productivity in that little field), and then trade the products of your expertise for necessities other people have created, all of you thereby gaining gains from trade.
So, contracts, which are basically enforceable agreements that facilitate trading with people who you do not personally know and therefore might not trust.
Let's start with a simple example. You want to buy some gewgaws from somebody. But you don't know them personally. The seller wants the money, you want their gewgaws, but because of the lack of trust (you don't know them!! what if they're scammers??) neither of you can benefit from gains from trade.
However, suppose both of you know of some entity that both of you trust. That entity can act as a trusted escrow. The entity provides you security: this enables the trade, allowing both of you to get gains from trade.
In Bitcoin-land, this can be implemented as a 2-of-3 multisignature. The three signatories in the multisgnature would be you, the gewgaw seller, and the escrow. You put the payment for the gewgaws into this 2-of-3 multisignature address.
Now, suppose it turns out neither of you are scammers (whaaaat!). You receive the gewgaws just fine and you're willing to pay up for them. Then you and the gewgaw seller just sign a transaction --- you and the gewgaw seller are 2, sufficient to trigger the 2-of-3 --- that spends from the 2-of-3 address to a singlesig the gewgaw seller wants (or whatever address the gewgaw seller wants).
But suppose some problem arises. The seller gave you gawgews instead of gewgaws. Or you decided to keep the gewgaws but not sign the transaction to release the funds to the seller. In either case, the escrow is notified, and if it can sign with you to refund the funds back to you (if the seller was a scammer) or it can sign with the seller to forward the funds to the seller (if you were a scammer).
Taproot helps with this: like mentioned above, it allows multisignature setups to produce only one signature, reducing blockchain space usage, and thus making contracts --- which require multiple people, by definition, you don't make contracts with yourself --- is made cheaper (which we hope enables more of these setups to happen for more gains from trade for everyone, also, moon and lambos).
(technology-wise, it's easier to make an n-of-n than a k-of-n, making a k-of-n would require a complex setup involving a long ritual with many communication rounds between the n participants, but an n-of-n can be done trivially with some moon math. You can, however, make what is effectively a 2-of-3 by using a three-branch SCRIPT: either 2-of-2 of you and seller, OR 2-of-2 of you and escrow, OR 2-of-2 of escrow and seller. Fortunately, Taproot adds a facility to embed a SCRIPT inside a public key, so you can have a 2-of-2 Taprooted address (between you and seller) with a SCRIPT branch that can instead be spent with 2-of-2 (you + escrow) OR 2-of-2 (seller + escrow), which implements the three-branched SCRIPT above. If neither of you are scammers (hopefully the common case) then you both sign using your keys and never have to contact the escrow, since you are just using the escrow public key without coordinating with them (because n-of-n is trivial but k-of-n requires setup with communication rounds), so in the "best case" where both of you are honest traders, you also get a privacy boost, in that the escrow never learns you have been trading on gewgaws, I mean ewww, gawgews are much better than gewgaws and therefore I now judge you for being a gewgaw enthusiast, you filthy gewgawer).

Taproot and Your Contracts, Part 2: Cryptographic Boogaloo

Now suppose you want to buy some data instead of things. For example, maybe you have some closed-source software in trial mode installed, and want to pay the developer for the full version. You want to pay for an activation code.
This can be done, today, by using an HTLC. The developer tells you the hash of the activation code. You pay to an HTLC, paying out to the developer if it reveals the preimage (the activation code), or refunding the money back to you after a pre-agreed timeout. If the developer claims the funds, it has to reveal the preimage, which is the activation code, and you can now activate your software. If the developer does not claim the funds by the timeout, you get refunded.
And you can do that, with HTLCs, today.
Of course, HTLCs do have problems:
Fortunately, with Schnorr (which is enabled by Taproot), we can now use the Scriptless Script constuction by Andrew Poelstra. This Scriptless Script allows a new construction, the PTLC or Pointlocked Timelocked Contract. Instead of hashes and preimages, just replace "hash" with "point" and "preimage" with "scalar".
Or as you might know them: "point" is really "public key" and "scalar" is really a "private key". What a PTLC does is that, given a particular public key, the pointlocked branch can be spent only if the spender reveals the private key of the given private key to you.
Another nice thing with PTLCs is that they are deniable. What appears onchain is just a single 2-of-2 signature between you and the developemanufacturer. It's like a magic trick. This signature has no special watermarks, it's a perfectly normal signature (the pledge). However, from this signature, plus some datta given to you by the developemanufacturer (known as the adaptor signature) you can derive the private key of a particular public key you both agree on (the turn). Anyone scraping the blockchain will just see signatures that look just like every other signature, and as long as nobody manages to hack you and get a copy of the adaptor signature or the private key, they cannot get the private key behind the public key (point) that the pointlocked branch needs (the prestige).
(Just to be clear, the public key you are getting the private key from, is distinct from the public key that the developemanufacturer will use for its funds. The activation key is different from the developer's onchain Bitcoin key, and it is the activation key whose private key you will be learning, not the developer's/manufacturer's onchain Bitcoin key).
So:
Taproot lets PTLCs exist onchain because they enable Schnorr, which is a requirement of PTLCs / Scriptless Script.
(technology-wise, take note that Scriptless Script works only for the "pointlocked" branch of the contract; you need normal Script, or a pre-signed nLockTimed transaction, for the "timelocked" branch. Since Taproot can embed a script, you can have the Taproot pubkey be a 2-of-2 to implement the Scriptless Script "pointlocked" branch, then have a hidden script that lets you recover the funds with an OP_CHECKLOCKTIMEVERIFY after the timeout if the seller does not claim the funds.)

Quantum Quibbles!

Now if you were really paying attention, you might have noticed this parenthetical:
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash...)
So wait, Taproot uses raw 32-byte public keys, and not public key hashes? Isn't that more quantum-vulnerable??
Well, in theory yes. In practice, they probably are not.
It's not that hashes can be broken by quantum computes --- they're still not. Instead, you have to look at how you spend from a P2WPKH/P2PKH pay-to-public-key-hash.
When you spend from a P2PKH / P2WPKH, you have to reveal the public key. Then Bitcoin hashes it and checks if this matches with the public-key-hash, and only then actually validates the signature for that public key.
So an unconfirmed transaction, floating in the mempools of nodes globally, will show, in plain sight for everyone to see, your public key.
(public keys should be public, that's why they're called public keys, LOL)
And if quantum computers are fast enough to be of concern, then they are probably fast enough that, in the several minutes to several hours from broadcast to confirmation, they have already cracked the public key that is openly broadcast with your transaction. The owner of the quantum computer can now replace your unconfirmed transaction with one that pays the funds to itself. Even if you did not opt-in RBF, miners are still incentivized to support RBF on RBF-disabled transactions.
So the extra hash is not as significant a protection against quantum computers as you might think. Instead, the extra hash-and-compare needed is just extra validation effort.
Further, if you have ever, in the past, spent from the address, then there exists already a transaction indelibly stored on the blockchain, openly displaying the public key from which quantum computers can derive the private key. So those are still vulnerable to quantum computers.
For the most part, the cryptographers behind Taproot (and Bitcoin Core) are of the opinion that quantum computers capable of cracking Bitcoin pubkeys are unlikely to appear within a decade or two.
So:
For now, the homomorphic and linear properties of elliptic curve cryptography provide a lot of benefits --- particularly the linearity property is what enables Scriptless Script and simple multisignature (i.e. multisignatures that are just 1 signature onchain). So it might be a good idea to take advantage of them now while we are still fairly safe against quantum computers. It seems likely that quantum-safe signature schemes are nonlinear (thus losing these advantages).

Summary

I Wanna Be The Taprooter!

So, do you want to help activate Taproot? Here's what you, mister sovereign Bitcoin HODLer, can do!