  We consider some generalization of the theory of quantum states, which is
based on the analysis of long standing problems and unsatisfactory situation
with the possible interpretations of quantum mechanics. We demonstrate that the
consideration of quantum states as sheaves can provide, in principle, more deep
understanding of some phenomena. The key ingredients of the proposed
construction are the families of sections of sheaves with values in the
category of the functional realizations of infinite-dimensional Hilbert spaces
with special (multiscale) filtration. Three different symmetries, kinematical
(on space-time), hidden/dynamical (on sections of sheaves), unified (on
filtration of the full scale of spaces) are generic objects generating the full
zoo of quantum phenomena (e.g., faster than light propagation).
