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The quantum dynamics of spin systems with uniform all-to-all interaction areoften studied in the totally symmetric space (TSS) of maximal total spin.However the TSS states are atypical in the full many-body Hilbert space. Inthis work, we explore several aspects of the all-to-all quantum dynamics awayfrom the TSS, and reveal surprising features of the "deep Hilbert space" (DHS).We study the out-of-time order correlator (OTOC) in the infinite-temperatureensemble of the full Hilbert space. We derive a phase-space representation ofthe DHS OTOC and show that the OTOC can grow super-exponentially in the large$N$ limit, due to the fast dynamics in an unbounded phase space (in finitesystems, we observe numerically that the super-exponential growth endsprecociously and gives way to a power-law one until saturation). By a similarmechanism, the Krylov complexity grows explosively. We also study theentanglement growth in a quantum quench from a DHS product state, i.e., one ofnon-aligned spins that resemble the DHS infinite-temperature ensemble withrespect to the statistics of the collective spins. Using a field-theoreticalmethod, We exactly calculate the entanglement entropy in the large $N$ limit.We show that, in the DHS, fast OTOC growth does not imply fast entanglementgrowth, in contrast to the Zurek-Paz relation derived in the TSS.

Surprises in the Deep Hilbert Space of all-to-all systems: From super-exponential scrambling to slow entanglement growth