Explore the intricacies of the Local Hamiltonian Problem in this 59-minute lecture by Sandy Irani from UC Irvine, presented at the Quantum Wave in Computing Boot Camp hosted by the Simons Institute. Delve into key concepts such as NP containment, ground state, locality, and adiabatic computation. Examine the proof techniques, propagation methods, and the role of clock states in quantum systems. Investigate energy density, Wang tiles, and their applications in quantum computing. Gain insights into common models, interaction graphs, and open quantum systems. Conclude with a discussion on future directions and potential applications of the Local Hamiltonian Problem in quantum computing research.