1. Proof of Work
Basically a proof of work is a piece of data that satifies certains requirements and is costly, time-consuming to produce but easily to be verified. Producing a proof of work can be a random process with low probability so that a lot of trial and error is required. The process of finding this proof of work is used to confirm transactions and "produce/validate" new blocks to the chain. The most widely used proof-of-work scheme is based on SHA-256 but some other hashing algorithms are used for proof-of-work: Scrypt, Blake-256, CryptoNight, HEFTY1, Quark, SHA-3, scrypt-jane, scrypt-n, and others.
Both Bitcoin and Ethereum uses proof of work. The proof of work in bitcoin is called "hashcash" and it consist in finding a sha2 hash with certain requirements (it must be lower tant a given target and begining with some zeroes).
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2. Proof of Stake
In the proof of stake, the validators (also called stakeholders) hold a stake in the system and is choosen to validate a new block.The chanche to be picked to validate a new block is proportional to the number of staked coins (coins set for this purpose). The coins staked cannot be used. Once a validator creates a block, the block should be put in the blockchain. How it is done vary in the different proof of stakes systems. Note that no coin creation exists in proof of stake. Instead, all the coins exist from day one, and validators are paid with the transaction fees.
Peercoin was the first coin to implement proof of stake.
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3. Proof of capacity
Basically the more free hard disk space you have, more likely you will be choosen to validate a block. This is because in this algotihm the space of the hard disk generates some data set know as 'plots' and more 'plots' a node has , more likely it will be choosen to validate the next block.
Burstcoin is the most known cryptocurrency to use a form of proof of capacity.
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4. Proof of Elapsed Time
In this algorithm, a leader ( or the 'validator') is elected based on a wait time received from a code running inside a trusted environment. In fact it is intended to work in a Trusted Execution Environment (TEE) such as Intel’s Software Guard Extensions (SGX). All validating or mining nodes have to run the TEE. Each validator requests a wait time from the code running inside the TEE. The validator with the shortest wait time wins the lottery and can become the leader, or the ones that can validate the block. When a validating node claims to be a leader and mines a block, it can also produce proof generated within the TEE that other nodes can easily verify.
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5. Proof of Activity
Proof of activity is a hybrid approach that combines both proof of work and proof of stake.
Basically In proof of activity miners races to solve a cryptographic puzzle, starting like a proof of work manner. In some implementation, blocks mined do not contain any transactions , so the winning block will only contain a header and the miner's reward address.
Then the system switches to proof of stake. Based on information in the header, a random group of validators is chosen for signing the new block. And as the proof of stake mecchanism, the more coins in the system a validator owns, the more likely he or she is to be chosen.
