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Many quantum systems can be described by the tensor product of small (e.g. two-dimensional) Hilbert spaces associated with atoms positioned in Rn, usually called “sites”. I will define a class of quantum states that can be reconstructed from local data corresponding to balls of radius r. Such states form a topological space, denoted by Bn for Bose systems and Fn for Fermi systems. (In the latter case, the Hilbert spaces are Z2-graded.) Only partial information is known about Bn and Fn, in particular, that they form ?-spectra. An analogous problem for so-called “free-fermion systems” is completely solved, and the corresponding spaces F(free)n are given by the KO spectrum. Interestingly, if we impose some symmetry described by an action of a compact group G on each site, then finding the fixed points in Bn and Fn is equivalent to finding the homotopy fixed points. That is not true in the free case.