Classically testing for the presence of anti-commuting operators on a quantum device is a critical tool underpinning recent progress in classical verification of quantum computation. While such tests can be based on cryptographic assumptions, known constructions rely on highly structured assumptions, e.g. trapdoor claw-free functions. In this work, we seek to explain this state of affairs by constructing strong cryptography from (certain forms of) classical tests of anti-commutation. In particular, we formulate the notion of a test of non-commutation (ToNC), an interactive protocol between a quantum prover and classical verifier in which the prover's final-round response is obtained by measuring one of two binary observables $P_0,P_1$ depending on the verifier's challenge bit $c$. We prove that, for a broad range of parameters, ToNC implies classical-communication key agreement (KA), and ToNC combined with one-way functions implies oblivious transfer (OT). Along the way, we develop tools for and provide the first known results on hardness amplification for post-quantum KA and OT, where communication is classical but adversaries may be quantum. In particular, we prove the following results of independent interest. - Post-quantum hard-core measure theorem: For any efficiently sampleable high-min-entropy distribution $D$ over pairs $(x,b)$ such that quantum circuits have advantage at most $δ$ in predicting $b$ from $x$, there exists a sub-distribution $M\preceq D$ of density $(1-δ)$ on which $b$ is nearly optimally quantum-hard to predict. - Post-quantum interactive XOR lemma: Given any classically-interactive protocol, if quantum adversaries have advantage at most $δ$ in guessing a private challenger bit $b$, then two sequential repetitions reduce the advantage for predicting the XOR of the challenger bits $b_1\oplus b_2$ to at most $δ^2+\rm{negl}(λ)$.

翻译：经典条件下对量子设备上反对易算子存在性的测试，是推动经典验证量子计算最新进展的关键工具。尽管此类测试可基于加密假设构建，但现有构造依赖于高度结构化的假设（如陷门爪形函数）。本文试图通过从（特定形式的）反对易的经典测试中构造强加密来解释这一现状。具体而言，我们提出了"非对易测试"（ToNC）概念——一种量子证明者与经典验证者之间的交互协议，其中证明者最终轮响应通过根据验证者挑战比特$c$测量两个二元可观测量$P_0,P_1$之一获得。我们证明在广泛参数范围内，ToNC蕴含经典通信密钥协商协议（KA），且ToNC结合单向函数可推导出不经意传输（OT）。在此过程中，我们为后量子KA与OT（通信为经典但攻击者可量子化）开发了难度放大工具并首次给出相关结论。特别地，我们证明了以下具有独立价值的结果： - 后量子硬核测度定理：对于任意可高效采样的高最小熵分布$D$（其样本为$(x,b)$且量子电路从$x$预测$b$的优势不超过$δ$），存在密度为$(1-δ)$的子分布$M\preceq D$使得$b$对量子攻击近乎最优困难地不可预测。 - 后量子交互XOR引理：对于任意经典交互协议，若量子敌手猜测私有挑战者比特$b$的优势不超过$δ$，则两次顺序重复可将预测挑战者比特异或值$b_1\oplus b_2$的优势降至$δ^2+\rm{negl}(λ)$。