  The relaxation of uniform quantum systems with finite-range interactions
after a quench is generically driven by the ballistic propagation of long-lived
quasi-particle excitations triggered by a sufficiently small quench. Here we
investigate the case of long-range ($1/r^{\alpha}$) interactions for
$d$-dimensional lattice spin models with uniaxial symmetry, and show that, in
the regime $d < \alpha < d+2$, the entanglement and correlation buildup is
radically altered by the existence of a non-linear dispersion relation of
quasi-particles, $\omega\sim k^{z<1}$, at small wave vectors, leading to a
divergence of the quasiparticle group velocity and super-ballistic propagation.
This translates in a super-linear growth of correlation fronts with time, and
sub-linear growth of relaxation times of subsystem observables with size, when
focusing on $k=0$ fluctuations. Yet the large dispersion in group velocities
leads to an extreme wavelength dependence of relaxation times of finite-$k$
fluctuations, with entanglement being susceptible to the longest of them. Our
predictions are directly relevant to current experiments probing the
nonequilibrium dynamics of trapped ions, or ultracold magnetic and Rydberg
atoms in optical lattices.
