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How can detector click probabilities respond to spatial rotations around afixed axis, in any possible physical theory? Here, we give a thoroughmathematical analysis of this question in terms of "rotation boxes", which areanalogous to the well-known notion of non-local boxes. We prove that quantumtheory admits the most general rotational correlations for spins 0, 1/2, and 1,but we describe a metrological game where beyond-quantum resources of spin 3/2outperform all quantum resources of the same spin. We prove a multitude offundamental results about these correlations, including an exact convexcharacterization of the spin-1 correlations, a Tsirelson-type inequality forspins 3/2 and higher, and a proof that the general spin-J correlations providean efficient outer SDP approximation to the quantum set. Furthermore, we reviewand consolidate earlier results that hint at a wealth of applications of thisformalism: a theory-agnostic semi-device-independent randomness generator, anexact characterization of the quantum (2,2,2)-Bell correlations in terms oflocal symmetries, and the derivation of multipartite Bell witnesses. Ourresults illuminate the foundational question of how space constrains thestructure of quantum theory, they build a bridge betweensemi-device-independent quantum information and spacetime physics, and theydemonstrate interesting relations to topics such as entanglement witnesses,spectrahedra, and orbitopes.

Spin-bounded correlations: rotation boxes within and beyond quantum theory