The Penrose inequality is a classical relation that specifies a lower bound on the total mass in terms of the area of certain trapped surfaces. We show that the inclusion of quantum effects violates this relation. We propose a quantum Penrose inequality: the lower bound on the total mass is set by a function of the generalized entropy of the light-sheet of any quantum marginally trapped surface. Hence, our conjecture relates quantum information in quantum gravity to the total energy of spacetime.