  We investigate the quantum Heisenberg model on the pyrochlore lattice for a
generic spin $S$ in the presence of nearest-neighbor $J_{1}$ and
second-nearest-neighbor $J_{2}$ exchange interactions. By employing the
pseudofermion functional renormalization group method, we find, for $S=1/2$ and
$S=1$, an extended quantum-spin-liquid phase centered around $J_{2}=0$, which
is shown to be robust against the introduction of breathing anisotropy. The
effects of temperature, quantum fluctuations, breathing anisotropies, and a
$J_{2}$ coupling on the nature of the scattering profile, and the pinch points,
in particular, are studied. For the magnetic phases of the $J_{1}$-$J_{2}$
model, quantum fluctuations are shown to renormalize phase boundaries compared
to the classical model and to modify the ordering wave vectors of spiral
magnetic states, while no new magnetic orders are stabilized.
