For each quantum system described by an operator algebraof physical quantities, we provide a (generalized) state space, notwithstanding the Kochen–Specker theorem. This quantum state space is the spectral presheaf. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheafgeometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.