  We theoretically study the dynamical phase diagram of the Dicke model in both
classical and quantum limits using large, experimentally relevant system sizes.
Our analysis elucidates that the model features dynamical critical points that
are distinct from previously investigated excited-state equilibrium
transitions. Moreover, our numerical calculations demonstrate that mean-field
features of the dynamics remain valid in the exact quantum dynamics, but we
also find that in regimes where quantum effects dominate signatures of the
dynamical phases and chaos can persist in purely quantum metrics such as
entanglement and correlations. Our predictions can be verified in current
quantum simulators of the Dicke model including arrays of trapped ions.
