Proof-of-Storage schemes allow a user to check if a storage provider is storing the outsourced data at the time of the challenge. How can we use PoS schemes to prove that some data was being stored throughout a period of time? A natural answer to this question is to require the user to repeatedly (e.g. every minute) send challenges to the storage provider. However, the communication complexity required in each interaction can be the bottleneck in systems such as Filecoin, where storage providers are required to submit their proofs to the blockchain network.
To address this question, we introduce a new proof, Proof-of-Spacetime, where a verifier can check if a prover is storing her/his outsourced data for a range of time. The intuition is to require the prover to (1) generate sequential Proofs-of-Storage (in our case Proof-of-Replication), as a way to determine time (2) recursively compose the executions to generate a short proof.
Section 3.3 of the Filecoin Paper provides the original introduction to Proof-of-Spacetime.