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Chaotic quantum systems with Lyapunov exponent $\lambda_\mathrm{L}$ obey anupper 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 andassociated non-trivial dynamics influence the interplay of quantum mechanicsand 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, ingeneral, make paramagnetic phase less and the replica symmetry-broken spinglass phase more chaotic. We show that the approach to the classical limitcould be non-trivial, with non-monotonic dependence of $\lambda_\mathrm{L}$ on$\hbar$ close to the dynamical glass transition temperature $T_d$. Our resultsin the classical limit ($\hbar\to 0$) naturally describe chaos in super-cooledliquid in structural glasses. We find a maximum in $\lambda_\mathrm{L}(T)$substantially above $T_d$, concomitant with the crossover from simple to slowglassy relaxation. We further show that $\lambda_\mathrm{L}\sim T^\alpha$, withthe exponent $\alpha$ varying between 2 and 1 from quantum to classical limit,at low temperatures in the spin glass phase.

Quantum to classical crossover in many-body chaos and scrambling from relaxation in a glass