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Proof of History
Relationship to consensus mechanisms
Most confusingly, a Proof of History (PoH) is more similar to a Verifiable Delay Function (VDF) than a Proof of Work or Proof of Stake consensus mechanism. The name unfortunately requires some historical context to understand. Proof of History was developed by Anatoly Yakovenko in November of 2017, roughly 6 months before we saw a paper using the term VDF. At that time, it was commonplace to publish new proofs of some desirable property used to build most any blockchain component. Some time shortly after, the crypto community began charting out all the different consensus mechanisms and because most of them started with "Proof of", the prefix became synonymous with a "consensus" suffix. Proof of History is not a consensus mechanism, but it is used to improve the performance of Solana's Proof of Stake consensus. It is also used to improve the performance of the replication and storage protocols. To minimize confusion, Solana may rebrand PoH to some flavor of the term VDF.
Relationship to VDFs
A desirable property of a VDF is that verification time is very fast. Solana's approach to verifying its delay function is proportional to the time it took to create it. Split over a 4000 core GPU, it is sufficiently fast for Solana's needs, but if you asked the authors the paper cited above, they might tell you (and have) that Solana's approach is algorithmically slow it shouldn't be called a VDF. We argue the term VDF should represent the category of verifiable delay functions and not just the subset with certain performance characteristics. Until that's resolved, Solana will likely continue using the term PoH for its application-specific VDF.
Another difference between PoH and VDFs used only for tracking duration, is that PoH's hash chain includes hashes of any data the application observed. That data is a double-edged sword. On one side, the data "proves history" - that the data most certainly existed before hashes after it. On the side, it means the application can manipulate the hash chain by changing when the data is hashed. The PoH chain therefore does not serve as a good source of randomness whereas a VDF without that data could. Solana's leader selection algorithm (TODO: add link), for example, is derived only from the VDF height and not its hash at that height.