Aaronson, Atia, and Susskind established that swapping quantum states $|\psi\rangle$ and $|\phi\rangle$ is computationally equivalent to distinguishing their superpositions $|\psi\rangle\pm|\phi\rangle$. We extend this to a general duality principle: manipulating quantum states in one basis is equivalent to extracting values in a complementary basis. Formally, for any group, implementing a unitary representation is equivalent to Fourier subspace extraction from its irreducible representations. Building on this duality principle, we present the applications: * Quantum money, representing verifiable but unclonable quantum states, and its stronger variant, quantum lightning, have resisted secure plain-model constructions. While (public-key) quantum money has been constructed securely only from the strong assumption of quantum-secure iO, quantum lightning has lacked such a construction, with past attempts using broken assumptions. We present the first secure quantum lightning construction based on a plausible cryptographic assumption by extending Zhandry's construction from Abelian to non-Abelian group actions, eliminating reliance on a black-box model. Our construction is realizable with symmetric group actions, including those implicit in the McEliece cryptosystem. * We give an alternative quantum lightning construction from one-way homomorphisms, with security holding under certain conditions. This scheme shows equivalence among four security notions: quantum lightning security, worst-case and average-case cloning security, and security against preparing a canonical state. * Quantum fire describes states that are clonable but not telegraphable: they cannot be efficiently encoded classically. These states "spread" like fire, but are viable only in coherent quantum form. The only prior construction required a unitary oracle; we propose the first candidate in the plain model.

翻译：Aaronson、Atia和Susskind证明了交换量子态$|\psi\rangle$与$|\phi\rangle$在计算上等价于区分它们的叠加态$|\psi\rangle\pm|\phi\rangle$。我们将此推广为一般对偶原理：在一个基中操纵量子态等价于在互补基中提取信息值。形式化地，对于任意群，实现其酉表示等价于从其不可约表示中进行傅里叶子空间提取。基于此对偶原理，我们提出以下应用：* 量子货币（代表可验证但不可克隆的量子态）及其强化变体量子闪电，长期以来未能实现安全的标准模型构造。虽然（公钥）量子货币仅能基于量子安全iO这一强假设实现安全构造，量子闪电则一直缺乏此类构造，过往尝试均基于已被攻破的假设。我们通过将Zhandry的构造从阿贝尔群作用推广至非阿贝尔群作用，首次基于可信密码学假设实现了安全的量子闪电构造，并消除了对黑盒模型的依赖。该构造可通过对称群作用实现，包括McEliece密码系统中隐含的群作用。* 我们提出基于单向同态的替代性量子闪电构造，其安全性在特定条件下成立。该方案证明了四种安全概念的等价性：量子闪电安全性、最坏情况与平均情况下的克隆安全性，以及抗规范态制备安全性。* 量子火焰描述可克隆但不可电报化的量子态：它们无法被高效地经典编码。这些态像火焰般"扩散"，但仅能以相干量子形式存在。此前唯一构造需要酉预言机；我们提出了标准模型下的首个候选方案。