Both the Proof of Replication and Proof of Spacetime processes in Filecoin use zk-SNARKs for compression.
zk-SNARKs stands for "Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge." You can think of them as hashes of computations. They let us prove that a proof has been done correctly without having to reveal the details of the proof itself or the underlying data on which it's based.
The process of creating Filecoin's zk-SNARKs is computationally expensive (slow), but the resulting end product is small and the verification process is very fast. Compared to the original proofs, zk-SNARKs are tiny, making them efficient to store in a blockchain. For example, a proof that would have taken up hundreds of kilobytes on the Filecoin chain can be compressed to just 192 bytes using a zk-SNARK.
As mentioned previously, everyone running a Filecoin node maintains an up-to-date version of the chain for verification purposes. Keeping each proof small with the assistance of zk-SNARKs minimizes the storage demands placed on each node in the Filecoin network, as well as the length of time it takes to verify a transaction.