Depth-efficient proofs of quantumness

A proof of quantumness is a type of challenge-response protocol in which a classical verifier can efficiently certify the quantum advantage of an untrusted prover. That is, a quantum prover can correctly answer the verifier's challenges and be accepted, while any polynomial-time classical prover will be rejected with high probability, based on plausible computational assumptions. To answer the verifier's challenges, existing proofs of quantumness typically require the quantum prover to perform a combination of polynomial-size quantum circuits and measurements. In this paper, we give two proof of quantumness constructions in which the prover need only perform constant-depth quantum circuits (and measurements) together with log-depth classical computation. Our first construction is a generic compiler that allows us to translate all existing proofs of quantumness into constant quantum depth versions. Our second construction is based around the learning with rounding problem, and yields circuits with shorter depth and requiring fewer qubits than the generic construction. In addition, the second construction also has some robustness against noise