Core Question
What makes addresses such as legacy, taproot, nested segwit and native segwit different from each other?
Legacy Addresses
Legacy addresses are the most basic of the 4. These addresses start with a 1 and use Base58 checks, meaning no letters and numbers like O, 0, I, etc. These were the original addresses introduced by Satoshi Nakamoto. These addresses were the most basic, just an address to send/receive your bitcoin. They were also very unoptimized, meaning that in today's world, the transaction fees are huge. You'll understand why as you keep reading through the other addresses.
Nested Segwit
Nested or wrapped segwit is exactly what it sounds like. It allowed users to use an exchange that didn't support the newer address type while bringing lower fees. It starts with a 3 and is about 25% cheaper in fees compared to the legacy address. It lowered fees by reducing how much data actually gets sent with your transaction to prove you own the funds. It essentially placed this heavy part of the fees into a sort of "extra block" attached to each bitcoin block. The network was then programmed to give a 75% discount on all fees from this extra block. That way, people would start using this address type. This address type also brought around optimized multi-sig (multi-sig existed before segwit, but not efficiently), meaning you could enable it so that a set number of wallets from a list had to sign a transaction before it could be broadcasted. It also was what allowed the Lightning network to be built, as you could now open and close lightning channels.
Native Segwit
This was also rolled out the same year as nested segwit (2017). However, instead of wrapping your address inside an old container, it used a brand new one starting with bc1q. It uses Bech32 coding, which means every letter is lowercase instead of both upper and lowercase. This makes for less errors and easier to read. There is also this check so that as you write the address in say your wallet, if you mistype a letter or number, the wallet will catch it and prevent you from sending the transaction. This address is about 11% cheaper than nested segwit and about 35% cheaper than legacy for the fees. This is due to the fact that it doesn't have to use the complex sort of bridging that nested segwit had in order to reuse the old legacy address format but with the segwit fee advantage of the "extra block". This type is supported by every exchange, hardware wallet, and software and is the go to and recommendation.
Taproot
This was released during a soft fork during November 2021. This address type uses bc1p and uses the same Bech32 generation as native segwit. However it now brings in Schnorr signature. Instead of the previous case where a multi-sig transaction was public and anyone could see what had to be done/what had been done to approve it (like wallets, number of signatures), Schnorr signature allowed all of that data to be aggregated into one signature. This not only lowered fees by reducing the number of pieces, but also improved privacy. It essentially makes every transaction look like a standard one wallet transaction, no matter what. This also means that a heavy multi-sig transaction costs as much as or a tiny bit more than a standard one wallet signature.
Due to Schnorr signatures, it also opened the door to NFTs on bitcoin, complex escrow and DeFi off chain, and making alternatives tokens directly on bitcoin. However, its usage is not universal. While pretty much everything in the world accepts sending to a taproot addresses, it is far less common to be able to receive from taproot, instead sticking to native segwit.
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