Pre-print, Working paper: Quantum cognition is a recent and rapidely growing field. In this paper we developan expected utility theory in a context of non-classical (quantum) uncertainty. We replace the classical state space with a Hilbert space which allows introducing the concept of quantum lottery. Within that framework we formulate sufficient and necessary axioms on preferences over quantum lotteries to establish a representation theorem. We show that demanding the consistency of choice behavior conditional on new information is equivalent to the von Neuman-Luders postulate applied to beliefs. In our context, dynamic consistency is shown not to secure Savage's Sure Thing Principle (in its dynamic version). Finally, we discuss the interpretation and value of our results for rationality and behavioral economics.