Uncloneable Quantum States Are Necessary as Proofs and Advice

Uncloneability is one of the most remarkable properties of quantum states that separates them from classical information. Over the years, this property has found numerous uses in cryptography - many tasks that are classically impossible become possible in quantum cryptography due to uncloneability. In this work, we examine the utility of uncloneable quantum states in complexity theory, specifically in the form of proofs and advice. We define and study languages that necessarily need uncloneable quantum proofs and advice. Specifically, we define strictly uncloneable versions of the classes QMA, BQP/qpoly and FEQP/qpoly (which is the class of relational problems solvable exactly with polynomial-sized quantum advice). Strictly uncloneable QMA is defined to be the class of languages in QMA that only have uncloneable proofs, i.e., given any family of candidate proof states, a polynomial-time cloning algorithm cannot act on it to produce states that are jointly usable by k separate polynomial-time verifiers, for arbitrary polynomial k. This is a stronger notion of uncloneable proofs and advice than those considered in previous works, which only required the existence of a single family of proof or advice states that are uncloneable. We show that in the quantum oracle model, there exist languages in strictly uncloneable QMA and strictly uncloneable BQP/qpoly. The language in strictly uncloneable QMA also gives a quantum oracle separation between QMA and the class cloneableQMA introduced by Nehoran and Zhandry (2024). We also show without using any oracles that the language, used by Aaronson, Buhrman and Kretschmer (2024) to separate FEQP/qpoly and FBQP/poly, is in strictly uncloneable FEQP/qpoly.