A proof of quantumness is a protocol through which a classical machine can test whether a purportedly quantum device, with comparable time and memory resources, is performing a computation that is impossible for classical computers. Existing approaches to provide proofs of quantumness depend on unproven assumptions about some task being impossible for machines of a particular model under certain resource restrictions. We study a setup where both devices have space bounds $\mathit{o}(\log \log n)$. Under such memory budgets, it has been unconditionally proven that probabilistic Turing machines are unable to solve certain computational problems. We formulate a new class of problems, and show that these problems are polynomial-time solvable for quantum machines, impossible for classical machines, and have the property that their solutions can be "proved" by a small-space quantum machine to a classical machine with the same space bound. These problems form the basis of our newly defined protocol, where the polynomial-time verifier's verdict about the tested machine's quantumness is not conditional on an unproven weakness assumption.

翻译：量子性证明是一种协议，通过该协议，经典机器可以测试一个声称具有量子能力的设备（在时间和内存资源相当的情况下）是否在执行经典计算机无法完成的计算。现有的量子性证明方法依赖于未经证明的假设，即某些任务对于特定模型下受资源限制的机器是不可能的。我们研究了一种设置，其中两种设备都具有空间界限 $\mathit{o}(\log \log n)$。在此类内存限制下，已无条件证明概率图灵机无法解决某些计算问题。我们定义了一类新问题，并证明这些问题对于量子机器是多项式时间可解的，对于经典机器是不可能的，并且其解可以由小空间量子机器向具有相同空间界限的经典机器"证明"。这些问题构成了我们新定义协议的基础，其中多项式时间验证器关于被测机器量子性的裁决不依赖于未经证明的弱点假设。