Explore quantum systems from a computational complexity perspective in this lecture from the Park City Mathematics Institute's Graduate Summer School program, where Sandy Irani delves into Quantum Hamiltonian complexity. Learn about computing fundamental properties of quantum systems, simulating particular systems over time, and understanding the complexity of ground states in both finite and infinite systems. Examine the computational resources required for these calculations while building on formal complexity-theoretic approaches rather than traditional heuristic methods. Access complementary lecture notes and problem session materials to reinforce understanding of these advanced quantum computing concepts, particularly focusing on the computational challenges in determining a system's lowest energy state or ground state.