In non-relativistic systems, the Lieb-Robinson Theorem imposes an emergent speed limit (independent of the relativistic limit set by c), establishing locality under unitary quantum dynamics and constraining the time needed to perform useful quantum tasks. We have extended the Lieb-Robinson Theorem to quantum dynamics with measurements. In contrast to the general expectation that measurements can arbitrarily violate spatial locality, we find at most an (M+1)-fold enhancement to the speed of quantum information, provided the outcomes of M local measurements are known; this holds even when classical communication is instantaneous. Our bound is asymptotically optimal, and saturated by existing measurement-based protocols (the "quantum repeater"). Our bound tightly constrain the resource requirements for quantum computation, error correction, teleportation, generating entangled resource states (Bell, GHZ, W, and spin-squeezed states), and preparing SPT states from short-range entangled states.

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