  We show that a broad class of quantum critical points can be stable against
locally correlated disorder even if they are unstable against uncorrelated
disorder. Although this result seemingly contradicts the Harris criterion, it
follows naturally from the absence of a random-mass term in the associated
order-parameter field theory. We illustrate the general concept with explicit
calculations for quantum spin-chain models. Instead of the infinite-randomness
physics induced by uncorrelated disorder, we find that weak locally correlated
disorder is irrelevant. For larger disorder, we find a line of critical points
with unusual properties such as an increase of the entanglement entropy with
the disorder strength. We also propose experimental realizations in the context
of quantum magnetism and cold-atom physics.
