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 verifiers challenges and be accepted, while any polynomial-time classical prover will be rejected with high probability, based on plausible computational assumptions. To answer the verifiers 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 show that all existing proofs of quantumness can be modified so that the prover need only perform constant-depth quantum circuits (and measurements) together with log-depth classical computation. We thus obtain depth-efficient proofs of quantumness whose soundness can be based on the same assumptions as existing schemes (factoring, discrete logarithm, learning with errors or ring learning with errors).
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