In non-relativistic quantum theories with short-range Hamiltonians, a
velocity $v$ can be chosen such that the influence of any local perturbation is
approximately confined to within a distance $r$ until a time $t \sim r/v$,
thereby defining a linear light cone and giving rise to an emergent notion of
locality. In systems with power-law ($1/r^{\alpha}$) interactions, when
$\alpha$ exceeds the dimension $D$, an analogous bound confines influences to
within a distance $r$ only until a time $t\sim(\alpha/v)\log r$, suggesting
that the velocity, as calculated from the slope of the light cone, may grow
exponentially in time. We rule out this possibility; light cones of power-law
interacting systems are algebraic for $\alpha>2D$, becoming linear as
$\alpha\rightarrow\infty$. Our results impose strong new constraints on the
growth of correlations and the production of entangled states in a variety of
rapidly emerging, long-range interacting atomic, molecular, and optical
systems.
