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.