We consider the role of quantum correlations in the efficient use of information by a predictive quantum system, generalizing a recently proposed classical measure of non-predictive information to the quantum regime. We show that, as a quantum system changes state, the non-predictive information held by another correlated quantum system is exactly equal to the extractable work that is lost from the second system. We use quantum discord to quantify the quantum contribution, and demonstrate the possibility of improved thermodynamic efficiency due to a negative "quantum part" of the lost work. We also give a thermodynamic interpretation to quantum discord, as the reduction in extractable work under an optimal classical approximation of a quantum memory.