A proof of quantumness is a type of challenge-response protocol in which a classical verifier can efficiently certify the $\textit{quantum advantage}$ 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 $\textit{constant-depth quantum circuits}$ (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 $\textit{learning with rounding}$ 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.

中文翻译：

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量子性的深度有效证明

量子性证明是一种挑战-响应协议，其中经典验证者可以有效地证明不受信任的证明者的 $\textit{quantum 优势}$。也就是说，量子证明者可以正确回答验证者的挑战并被接受，而基于合理的计算假设，任何多项式时间的经典证明者都将以高概率被拒绝。为了回答验证者的挑战，现有的量子性证明通常需要量子证明者执行多项式大小的量子电路和测量的组合。
在本文中，我们给出了两个量子结构证明，其中证明者只需要执行 $\textit{恒定深度量子电路}$（和测量）以及对数深度经典计算。我们的第一个构造是一个通用编译器，它允许我们将所有现有的量子性证明转换为恒定的量子深度版本。我们的第二个构造基于 $\textit{learning with rounding}$ 问题，并且产生的电路比通用构造具有更短的深度和更少的量子比特。此外，第二种结构也具有一定的抗噪声鲁棒性。