  Chaotic quantum systems with Lyapunov exponent $\lambda_\mathrm{L}$ obey an
upper bound $\lambda_\mathrm{L}\leq 2\pi k_\mathrm{B}T/\hbar$ at temperature
$T$, implying a divergence of the bound in the classical limit $\hbar\to 0$.
Following this trend, does a quantum system necessarily become `more chaotic'
when quantum fluctuations are reduced? Moreover, how do symmetry breaking and
associated non-trivial dynamics influence the interplay of quantum mechanics
and chaos? We explore these questions by computing
$\lambda_\mathrm{L}(\hbar,T)$ in the quantum spherical $p$-spin glass model,
where $\hbar$ can be continuously varied. We find that quantum fluctuations, in
general, make paramagnetic phase less and the replica symmetry-broken spin
glass phase more chaotic. We show that the approach to the classical limit
could be non-trivial, with non-monotonic dependence of $\lambda_\mathrm{L}$ on
$\hbar$ close to the dynamical glass transition temperature $T_d$. Our results
in the classical limit ($\hbar\to 0$) naturally describe chaos in super-cooled
liquid in structural glasses. We find a maximum in $\lambda_\mathrm{L}(T)$
substantially above $T_d$, concomitant with the crossover from simple to slow
glassy relaxation. We further show that $\lambda_\mathrm{L}\sim T^\alpha$, with
the exponent $\alpha$ varying between 2 and 1 from quantum to classical limit,
at low temperatures in the spin glass phase.
