The non-equilibrium response of a quantum many-body system defines its fundamental transport properties and how initially localized quantum information spreads. However, for long-range-interacting quantum systems little is known. We address this issue by analyzing a local quantum quench in the long-range Ising model in a transverse field, where interactions decay as a variable power-law with distance $propto r^{-alpha}$, $alpha>0$. Using complementary numerical and analytical techniques, we identify three dynamical regimes: short-range-like with an emerging light cone for $alpha>2$; weakly long-range for $1<alpha<2$ without a clear light cone but with a finite propagation speed of almost all excitations; and fully non-local for $alpha<1$ with instantaneous transmission of correlations. This last regime breaks generalized Lieb--Robinson bounds and thus locality. Numerical calculation of the entanglement spectrum demonstrates that the usual picture of propagating quasi-particles remains valid, allowing an intuitive interpretation of our findings via divergences of quasi-particle velocities. Our results may be tested in state-of-the-art trapped-ion experiments.
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