  The quantum dynamics of spin systems with uniform all-to-all interaction are
often studied in the totally symmetric space (TSS) of maximal total spin.
However the TSS states are atypical in the full many-body Hilbert space. In
this work, we explore several aspects of the all-to-all quantum dynamics away
from the TSS, and reveal surprising features of the "deep Hilbert space" (DHS).
We study the out-of-time order correlator (OTOC) in the infinite-temperature
ensemble of the full Hilbert space. We derive a phase-space representation of
the 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 finite
systems, we observe numerically that the super-exponential growth ends
precociously and gives way to a power-law one until saturation). By a similar
mechanism, the Krylov complexity grows explosively. We also study the
entanglement growth in a quantum quench from a DHS product state, i.e., one of
non-aligned spins that resemble the DHS infinite-temperature ensemble with
respect to the statistics of the collective spins. Using a field-theoretical
method, We exactly calculate the entanglement entropy in the large $N$ limit.
We show that, in the DHS, fast OTOC growth does not imply fast entanglement
growth, in contrast to the Zurek-Paz relation derived in the TSS.
