The famous no-cloning principle has been shown recently to enable a number of uncloneable functionalities. Here we address for the first time unkeyed quantum uncloneablity, via the study of a complexity-theoretic tool that enables a computation, but that is natively unkeyed: quantum advice. Remarkably, this is an application of the no-cloning principle in a context where the quantum states of interest are not chosen by a random process. We show the unconditional existence of promise problems admitting uncloneable quantum advice, and the existence of languages with uncloneable advice, assuming the feasibility of quantum copy-protecting certain functions. Along the way, we note that state complexity classes, introduced by Rosenthal and Yuen (ITCS 2022) - which concern the computational difficulty of synthesizing sequences of quantum states - can be naturally generalized to obtain state cloning complexity classes. We make initial observations on these classes, notably obtaining a result analogous to the existence of undecidable problems. Our proof technique establishes the existence of ingenerable sequences of finite bit strings - essentially meaning that they cannot be generated by any uniform circuit family. We then prove a generic result showing that the difficulty of accomplishing a computational task on uniformly random inputs implies its difficulty on any fixed, ingenerable sequence. We use this result to derandomize quantum cryptographic games that relate to cloning, and then incorporate a result of Kundu and Tan (arXiv 2022) to obtain uncloneable advice. Applying this two-step process to a monogamy-of-entanglement game yields a promise problem with uncloneable advice, and applying it to the quantum copy-protection of pseudorandom functions with super-logarithmic output lengths yields a language with uncloneable advice.

翻译：著名的不可克隆原理近期已被证明能够实现多种不可克隆功能。本文首次通过研究一种实现计算但本质上无密钥的复杂性理论工具——量子建议，来探讨无密钥的量子不可克隆性。值得注意的是，这是不可克隆原理在量子态并非由随机过程选择的场景下的应用。我们证明了承诺问题无条件存在可接受的不可克隆量子建议，并假设某些函数具备量子复制保护能力时，语言类也可存在不可克隆建议。在此过程中，我们注意到由Rosenthal和Yuen（ITCS 2022）提出的态复杂性类——关注量子态序列合成的计算难度——可自然推广至态克隆复杂性类。我们对此类进行了初步观察，特别得到了与不可判定问题存在性类似的结论。我们的证明技术确立了有限比特串序列的不可生成性——本质上意味着它们无法被任何一致电路族生成。随后，我们证明了一个通用结论：在均匀随机输入上完成计算任务的困难性意味着其在任何固定不可生成序列上的困难性。利用该结论对与克隆相关的量子密码博弈进行去随机化，并结合Kundu和Tan（arXiv 2022）的结果获得不可克隆建议。将这一两步流程应用于纠缠负性博弈，得到具有不可克隆建议的承诺问题；应用于超对数输出长度的伪随机函数量子复制保护，得到具有不可克隆建议的语言类。