  Quantum cognition in decision-making is a recent and rapidely growing field.
In this paper we develop an 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 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-L\"uders postulate applied to beliefs. A dynamically consistent
quantum-like agent may violate dynamic recursive consistency, however. This
feature suggests interesting applications in behavioral economics as we
illustrate in an example of persuasion.
