A typical problem in physics is to describe the ground state or time evolution of a system of N particles, each of which has 2 different states. The Hilbert space for this system has dimension 2^N, and hence a brute force approach to finding the ground state is completely impractical for even modest N. In these talks I will discuss recent results on both of these problems, including "easiness results" (such as the result that gapped one dimensional quantum systems are in NP) and "hardness results" (such as the difficulty of determining consistency of local density matrices).