Proof of space

Proof of space (PoS) is a means of showing that one has a legitimate interest in a service (such as sending an email) by allocating a non-trivial amount of memory or disk space to solve a challenge presented by the service provider. The concept was formulated in 2013 by Dziembowski "et al.", and (with a different formulation) by Ateniese "et al.". 
Proofs of space are very similar to proofs of work (PoW), except that instead of computation, storage is used. Proof-of-space is different from memory-hard functions in that the bottleneck is not in the number of memory access events, but in the amount of memory required.
After the release of Bitcoin, alternatives to its PoW mining mechanism were researched and PoS was studied in the context of cryptocurrencies. Proofs of space are seen as a fairer and greener alternative by blockchain enthusiasts due to the general-purpose nature of storage and the lower energy cost required by storage, but have been criticized for increasing demand for storage. Several theoretical and practical implementations of PoS have been released and discussed, such as SpaceMint and Burstcoin.
Concept description.
A proof-of-space is a piece of data that a prover sends to a verifier to prove that the prover has reserved a certain amount of space. For practicality, the verification process needs to be efficient, namely, consume a small amount of space and time. For soundness, it should be hard for the prover to pass the verification if it does not actually reserve the claimed amount of space. One way of implementing PoS is by using hard-to-pebble graphs. The verifier asks the prover to build a labeling of a hard-to-pebble graph. The prover commits to the labeling. The verifier then asks the prover to open several random locations in the commitment.
Proof of storage.
A proof of storage (also proof of retrievability, proof of data possession) is related to a proof-of-space, but instead of showing that space is available for solving a puzzle, the prover shows that space is actually used to store a piece of data correctly at the time of proof.
Cryptocurrencies intended to assign value to store data use some form of this system; examples include Storj, Sia, Filecoin, and Chia.
Proof of capacity.
A proof of capacity is a system where miners are allowed to pre-calculate ("plot") PoW functions and store them onto the HDD. It is generally only used by Burstcoin. 
Proof of space-time.
A proof of space-time is a proof that shows the prover has spent an amount of time keeping the reserved space unchanged. Its creators reason that the cost of storage is inextricably linked not only to its capacity, but to the time in which that capacity is used. It is related to a proof-of-storage, although the Moran-Orlov construction also allows a tradeoff between space and time.
Uses.
Proofs of space could be used as an alternative to proofs of work in the traditional client puzzle applications such as anti-spam measures and denial of service attack prevention. Proof-of-Space has also been used for malware detection, by determining whether the L1 cache of a processor is empty (e.g., has enough space to evaluate the PoS routine without cache misses) or contains a routine that resisted being evicted.
Proofs of space have been developed further in several concept papers and in one live cryptocurrency implementation.
Implementations.
Burstcoin.
Proof of capacity consumes disk space rather than computing resources to mine a block. Unlike PoW, where the miners keep changing the block header and hash to find the solution, the Proof of capacity implementation in Burstcoin generates random solutions, also called plots, using the Shabal cryptographic algorithm in advance and stores it on hard drives. This stage is called plotting and it may take days or even weeks depending on the storage capacity of the drive. In the next stage - mining, miners match their solutions to the most recent puzzle and the node with the fastest solution gets to mine the next block.
Concepts.
SpaceMint.
In 2015, a paper proposed a cryptocurrency called SpaceMint. It attempts to solve some of the practical design problems associated with the pebbling-based PoS schemes. In using PoS for decentralized cryptocurrency, the protocol has to be adapted to work in a non-interactive protocol since each individual in the network has to behave as a verifier.
Chia.
In 2018, a proposed cryptocurrency Chia presented two papers presenting a new protocol based on proof of space and proof of time.
In February of 2021, Chia published a white paper outlining its business and has since launched its mainnet and Chia coin (XCH) using the Proof of Space Time concept.
The involvement of storage media (hard disk and solid-state drives) as the cryptocurrency's mining medium has raised concerns over potential price surges and shortage of high-capacity storage devices.