  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.
