On its face, a quantum state of n entangled particles contains anamount of information exponential in n. In this talk, however, I'll survey recent results showing that, in many senses relevant for computation, prediction, and learning, quantum states don't behave as if they contained an exponential amount of information. Specifically, I'll discuss the limitations of "quantum advice states," the approximate "learnability" of quantum states from random measurement results, and a recently-discovered simulation of arbitrary quantum states by the ground states of local Hamiltonians. Some of the talk represents joint work with Andrew Drucker and Eyal Dechter.