Model-independent tests of quantum theory, such as Bell inequality violations, focus on static properties of the theory and do not rely on its dynamical aspects. Hence, this core part of quantum theory remains largely untested in a model-independent way. We show that the structure of quantum time evolution can be examined by testing the consistency between the state space and the dynamical laws. Considering a class of generalized phase-space theories, we find that certain quantum states can only be part of a theory if the time evolution is quantum: their evolution in anharmonic potentials would otherwise result in unphysical, negative marginal distributions. Furthermore, we demonstrate that the existence of the first excited state of a quantum harmonic oscillator is already sufficient to single out the Moyal bracket (quantum dynamics) to leading order in
ℏ2. We outline how theory-independent Wigner tomography of this state can be used to empirically exclude nonquantum dynamical theories.
Most recently 't Hooft has postulated (G. 't Hooft, Class. Quantum Grav.16, 3263 (1999)) that quantum states at the "atomic scale" can be understood as equivalence classes of primordial states governed by a dissipative deterministic theory underlying quantum theory at the "Planck scale". Defining invariant subspaces clearly for primordial states according to a given evolution, we mathematically reformulate 't Hooft's theory as a quotient space construction with the time-reversible evolution operator induced naturally. With this observation and some analysis, 't Hooft's theory is generalized beyond his case where the evolution at the "Planck scale" is a power of a one-time-step evolution or the time is discrete.