  A modern version of the idea that the area of event horizons gives $4G$ times
an entropy is the Hubeny-Rangamani Causal Holographic Information (CHI)
proposal for holographic field theories. Given a region $R$ of a holographic
QFTs, CHI computes $A/4G$ on a certain cut of an event horizon in the
gravitational dual. The result is naturally interpreted as a coarse-grained
entropy for the QFT. CHI is known to be finitely greater than the fine-grained
Hubeny-Rangamani-Takayanagi (HRT) entropy when $\partial R$ lies on a Killing
horizon of the QFT spacetime, and in this context satisfies other non-trivial
properties expected of an entropy. Here we present evidence that it also
satisfies the quantum null energy condition (QNEC), which bounds the second
derivative of the entropy of a quantum field theory on one side of a
non-expanding null surface by the flux of stress-energy across the surface. In
particular, we show CHI to satisfy the QNEC in 1+1 holographic CFTs when
evaluated in states dual to conical defects in AdS$_3$. This surprising result
further supports the idea that CHI defines a useful notion of coarse-grained
holographic entropy, and suggests unprecedented bounds on the rate at which
bulk horizon generators emerge from a caustic. To supplement our motivation, we
include an appendix deriving a corresponding coarse-grained generalized second
law for 1+1 holographic CFTs perturbatively coupled to dilaton gravity.
