Aaronson, Atia, and Susskind established that swapping quantum states |ψ⟩ and |ϕ⟩ is computationally equivalent to distinguishing their superpositions |ψ⟩ ± |ϕ⟩. 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 computationally equivalent to Fourier subspace extraction from its irreducible representations.

Building on this duality principle, we present the following 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 in the plain model 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 with a classical oracle.

Based on joint work with John Bostanci and Mark Zhandry.

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