Chaotic quantum systems with Lyapunov exponent obey an upper bound at temperature , implying a divergence of the bound in the classical limit . Following this trend, does a quantum system necessarily become “more chaotic” when quantum fluctuations are reduced? Moreover, how do symmetry breaking and associated nontrivial dynamics influence the interplay of quantum mechanics and chaos? We explore these questions by computing in the quantum spherical -spin glass model, where 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 nontrivial, with nonmonotonic dependence of on close to the dynamical glass transition temperature . Our results in the classical limit () naturally describe chaos in supercooled liquid in structural glasses. We find a maximum in substantially above , concomitant with the crossover from simple to slow glassy relaxation. We further show that , with the exponent varying between 2 and 1 from quantum to classical limit, at low temperatures in the spin glass phase.