  While classical chaos has been successfully characterized with consistent
theories and intuitive techniques, such as with the use of Lyapunov exponents,
quantum chaos is still poorly understood, as well as its relation with
multi-partite entanglement and information scrambling. We consider a benchmark
system, the kicked top model, which displays chaotic behaviour in the classical
version, and proceed to characterize the quantum case with a thorough diagnosis
of the growth of chaos and entanglement in time. As a novel tool for the
characterization of quantum chaos, we introduce for this scope the
quasi-probability distribution behind the out-of-time-ordered correlator
(OTOC). We calculate the cumulative nonclassicality of this distribution, which
has already been shown to outperform the simple use of OTOC as a probe to
distinguish between integrable and nonintegrable Hamiltonians. To provide a
thorough comparative analysis, we contrast the behavior of the nonclassicality
with entanglement measures, such as the tripartite mutual information of the
Hamiltonian as well as the entanglement entropy. We find that systems whose
initial states would lie in the "sea of chaos" in the classical kicked-top
model, exhibit, as they evolve in time, characteristics associated with chaotic
behavior and entanglement production in closed quantum systems. We corroborate
this indication by capturing it with this novel OTOC-based measure.
