  Many-body cavity QED experiments are established platforms to tailor and
control the collective responses of ensembles of atoms, interacting through one
or more common photonic modes. The rich diversity of dynamical phases they can
host, calls for a unified framework. Here we commence this program by showing
that a cavity QED simulator assembled from $N$-levels bosonic atoms, can
reproduce and extend the possible dynamical responses of collective observables
occurring after a quench. Specifically, by initializing the atoms in classical
or quantum states, or by leveraging intra-levels quantum correlations, we craft
on demand the entire synchronization/desynchronization dynamical crossover of
an exchange model for $SU(N)$ spins. We quantitatively predict the onset of
different dynamical responses by combining the Liouville-Arnold theorem on
classical integrability with an ansatz for reducing the collective evolution to
an effective few-body dynamics. Among them, we discover a synchronized chaotic
phase induced by quantum correlations and associated to a first order
non-equilibrium transition in the Lyapunov exponent of collective atomic
dynamics. Our outreach includes extensions to other spin-exchange quantum
simulators and a universal conjecture for the dynamical reduction of
non-integrable all-to-all interacting systems.
