Alice performs a very complex calculation with algorithm X with astronomical calculation costs at time t. Is it possible for Bob to verify that the calculation was done with algorithm X at time t, without costing him anything? Could homomorphic coding be useful for this, and if so, how?

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Blockchain pretty much does this.

Verifying that it happened at some timepoint can only be done approximately. For example, proof-of-work hashes in a blockchain indicates the hash in one block was most certainly not computed before the preceeding block and there has very likely passed some time since relative to the amount of proof-of-work hashes that follows.

But it’s still an approximation, because it requires that you make assumptions about computation speeds and resource availability, since timestamps can be falsified if you create a private chain.

Proving that a certain arbitrary computation (not just for proof-of-work algorithms!) was made at some point is however doable with far higher certainty with zero-knowledge proofs. Keep in mind that computing a zero-knowledge proof itself adds overhead, making the computation even more costly.