Publicly verifiable delegation is a well-known problem involving a user who wishes to outsource a resource-intensive computational task to a more powerful but potentially untrusted server such that any other party is able to efficiently check the veracity of the computation's result. This problem has been extensively studied in the classical domain where the user and server are both non-quantum machines. However, the problem becomes more challenging when the classical user wants to delegate a quantum circuit to a single prover with quantum-computing capabilities. Previous solutions have resorted to using impractical or non-standard cryptographic solutions (e.g. indistinguishability obfuscation) to achieve this requirement. In this work, we relax the requirement to have time-delayed publicly verifiable proofs, where the verification key is made known to the public only when the computation (and its proof) are guaranteed to have been completed. We propose a practical non-interactive scheme leveraging commitment schemes and time-lock puzzles, which can be efficiently realized through well-established and standard post-quantum assumptions. The main idea of our technique lies in using time-lock puzzles to compile a 2-round privately verifiable scheme into a non-interactive publicly verifiable scheme with timestamped proofs, outsourcing not only the quantum computation but the puzzle solving as well. Security is proven in the quantum random oracle model with a common reference string (CRS).

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