  We investigate the conductance through and the spectrum of ballistic chaotic
quantum dots attached to two s-wave superconductors, as a function of the phase
difference $\phi$ between the two order parameters. A combination of analytical
techniques -- random matrix theory, Nazarov's circuit theory and the
trajectory-based semiclassical theory -- allows us to explore the
quantum-to-classical crossover in detail. When the superconductors are not
phase-biased, $\phi=0$, we recover known results that the spectrum of the
quantum dot exhibits an excitation gap, while the conductance across two normal
leads carrying $N_{\rm N}$ channels and connected to the dot via tunnel
contacts of transparency $\Gamma_{\rm N}$ is $\propto \Gamma_{\rm N}^2 N_{\rm
N}$. In contrast, when $\phi=\pi$, the excitation gap closes and the
conductance becomes $G \propto \Gamma_{\rm N} N_{\rm N}$ in the universal
regime. For $\Gamma_{\rm N} \ll 1$, we observe an order-of-magnitude
enhancement of the conductance towards $G \propto N_{\rm N}$ in the
short-wavelength limit. We relate this enhancement to resonant tunneling
through a macroscopic number of levels close to the Fermi energy. Our
predictions are corroborated by numerical simulations.
