  We prove novel speed limits on the growth of entanglement, equal-time
correlators, and spacelike Wilson loops in spatially uniform time-evolving
states in strongly coupled CFTs with holographic duals. These bounds can also
be viewed as quantum weak energy conditions. Several of the speed limits are
valid for regions of arbitrary size and with multiple connected components, and
our findings imply new bounds on the effective entanglement velocity of small
subregions. In 2d CFT, our results prove a conjecture by Liu and Suh for a
large class of states. We also bound spatial derivatives of entanglement and
correlators. Key to our findings is a momentum-entanglement correspondence,
showing that entanglement growth is computed by the momentum crossing the HRT
surface. In our setup, we prove a number of general features of
boundary-anchored extremal surfaces, such as a sharp bound on the smallest
radius that a surface can probe, and that the tips of extremal surfaces cannot
lie in trapped regions. Our methods rely on novel global GR techniques,
including a delicate interplay between Lorentzian and Riemannian Hawking
masses. While our proofs assume the dominant energy condition in the bulk, we
provide numerical evidence that our bounds are true under less restrictive
assumptions.
