  In many-body quantum systems with spatially local interactions, quantum
information propagates with a finite velocity, reminiscent of the ``light cone"
of relativity. In systems with long-range interactions which decay with
distance $r$ as $1/r^\alpha$, however, there are multiple light cones which
control different information theoretic tasks. We show an optimal (up to
logarithms) ``Frobenius light cone" obeying $t\sim r^{\min(\alpha-1,1)}$ for
$\alpha>1$ in one-dimensional power-law interacting systems with finite local
dimension: this controls, among other physical properties, the butterfly
velocity characterizing many-body chaos and operator growth. We construct an
explicit random Hamiltonian protocol that saturates the bound and settles the
optimal Frobenius light cone in one dimension. We partially extend our
constraints on the Frobenius light cone to a several operator $p$-norms, and
show that Lieb-Robinson bounds can be saturated in at most an exponentially
small $e^{-\Omega(r)}$ fraction of the many-body Hilbert space.
