Explore the quantum analog of NP-completeness in this Richard M. Karp Distinguished Lecture delivered by Sandy Irani from UC Irvine. Delve into the local Hamiltonian problem, the cornerstone of quantum Hamiltonian complexity, and its parallels with classical constraint satisfaction. Examine how key aspects of NP-completeness translate to the quantum setting, including search-to-decision, dichotomy theorems, unique solutions, and approximation. Gain insights into potential strategies for addressing QMA-hardness in quantum applications. Learn from Irani's expertise spanning online algorithms, scheduling, resource allocation, and quantum computation, with a focus on quantum complexity theory. This lecture, part of a series honoring Richard M. Karp's contributions to theoretical computer science, offers a comprehensive overview of quantum constraint satisfaction for a broad scientific audience.