The quantum null energy condition (QNEC) is a conjectured bound on components
$(T_{kk} = T_{ab} k^a k^b$) of the stress tensor along a null vector $k^a$ at a
point $p$ in terms of a second $k$-derivative of the von Neumann entropy $S$ on
one side of a null congruence $N$ through $p$ generated by $k^a$. The
conjecture has been established for super-renormalizeable field theories at
points $p$ that lie on a bifurcate Killing horizon with null tangent $k^a$ and
for large-N holographic theories on flat space. While the Koeller-Leichenauer
holographic argument clearly yields an inequality for general $(p,k^a)$, more
conditions are generally required for this inequality to be a useful QNEC. For
$d\le 3$, for arbitrary backgroud metric satisfying the null convergence
condition $R_{ab} k^a k^b \ge 0$, we show that the QNEC is naturally finite and
independent of renormalization scheme when the expansion $\theta$ and shear
$\sigma_{ab}$ of $N$ at point $p$ satisfy $\theta |_p= \dot{\theta}|_p =0$,
$\sigma_{ab}|_p=0$. This is consistent with the original QNEC conjecture. But
for $d=4,5$ more conditions are required. In particular, we also require the
vanishing of additional derivatives and a dominant energy condition. In the
above cases the holographic argument does indeed yield a finite QNEC, though
for $d\ge6$ we argue these properties to fail even for weakly isolated horizons
(where all derivatives of $\theta, \sigma_{ab}$ vanish) that also satisfy a
dominant energy condition. On the positive side, a corrollary to our work is
that, when coupled to Einstein-Hilbert gravity, $d \le 3$ holographic theories
at large $N$ satisfy the generalized second law (GSL) of thermodynamics at
leading order in Newton's constant $G$. This is the first GSL proof which does
not require the quantum fields to be perturbations to a Killing horizon.
