How does a miner calculate spacetime?
At a high level miners run a repeated proof of replication. To do this they must (1) have some data stored and “sealed” in a particular way and (2) sample a random challenge from the chain to start running proofs. After each proof is generated from a challenge it becomes the input of the next proof.
Proofs cannot (except with very low probability) be generated without access to the sealed data so this is a “proof of space”, multiple steps of this chain of proofs cannot be parallelized (except with very low probability) so this is a “proof of time”.
At the end the miner calculates another proof, a proof that the miner calculated a chain of proofs. This is where snarks are used. Finally the miner can submit the output of this “meta-proof” to the chain and other miners can use it to efficiently verify that the submitting miner has stored the sealed data over time.
I am not a proofs wizard, and PoST is an active area of research/dev
At a high level miners run a repeated proof of replication.
Small comment: we perform an initial proof of replication, then repeated proofs of space-time. The first one is the most expensive and establishes that the data has been transformed to a unique replica. Subsequent proofs show that this replica is being stored continuously.