While quantum position verification aims to certify a prover's location using quantum information, existing security definitions only guarantee that part of the successful adversarial party is in the claimed location.This leaves open the possibility that a distributed team of adversaries can jointly simulate a prover in a way that defeats the intended meaning of `being at a location' in position-based cryptography. We introduce stronger notions of position verification that we call quantum localization, which requires that there is a specified, unclonable state at the verified spacetime point-and that this state can be found nowhere else.We show that quantum localization leads naturally to a meaningful notion of trajectory verification, in which quantum information is verifiably tracked through space and time.We construct quantum localization and trajectory verification protocols using quantum anchor states, which generalize coset states from unclonable cryptography.The security of our schemes is proven in the classical oracle (i.e. ideal obfuscation) model, which can be heuristically instantiated in the plain model using post-quantum indistinguishability obfuscation. We also introduce and instantiate the concept of functionality localization, which guarantees that the adversary has the ability to compute a secret function at the verified spacetime point, and this function cannot be computed anywhere else.This raises the intriguing possibility of localizing computational capabilities in space and time. More broadly, we believe our notions of quantum localization and subsequent feasibility results provide stronger foundations for position-based cryptography.At a conceptual level, our results also explore the tight link between two fundamental notions in physics-location and information-thus serving as a natural foundation for cryptographic capabilities tied to physical spacetime.

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