In quantum field theory it is possible to create negative local energy densities. This would violate the Generalized Second Law (GSL) unless there is some sort of energy condition requiring the negative energy to be counterbalanced by positive energy. TO explore what this energy condition is, I will assume that the GSL holds in semiclassical gravity for all future causal horizons. From CPT symmetry it follows that the time-reverse of the GSL, properly understood, holds for all past causal horizons. These two conditions together then imply that the Averaged Null Energy Condition (ANEC) holds on any null line, i.e. a complete achronal lightlike null geodesic. In curved spacetimes, the ANEC can be violated on general geodesics. But even if the ANEC only holds on null lines, theorems by Sorkin, Penrose and Woolgar, and by Graham and Olum imply that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. These results can thus be seen as consequences of the GSL. However, these theorems don't apply when gravitational fluctuations are taken into account. In that case, the GSL argument suggests a modification to the ANEC which may make these theorems applicable to perturbative quantum gravity.