  Although highly successful, the truncated Wigner approximation (TWA) leaves
out many-body quantum interference between mean-field Gross-Pitaevskii
solutions as well as other quantum effects, and is therefore essentially
classical. Turned around, this implies that if a system's quantum properties
deviate from TWA, they must be exhibiting some quantum phenomenon, such as
localization, diffraction, or tunneling. Here, we consider in detail a
particular interference effect arising from discrete symmetries, which can lead
to a significant enhancement of quantum observables with respect to the TWA
prediction, and derive an augmented version of the TWA in order to incorporate
them. Using the Bose-Hubbard model for illustration, we further show strong
evidence for the presence of dynamical localization due to remaining
differences between the TWA predictions and quantum results.
