  Going beyond the currently investigated regimes in experiments on quantum
transport of ultracold atoms in disordered potentials, we predict a crossover
between regular and quantum-chaotic dynamics when varying the strength of
disorder. Our spectral approach is based on the Bose-Hubbard model describing
interacting atoms in deep random potentials. The predicted crossover from
localized to diffusive dynamics depends on the simultaneous presence of
interactions and disorder, and can be verified in the laboratory by monitoring
the evolution of typical experimental initial states.
