This talk is about two Hamiltonian Complexity questions. First, how hard is it to compute the ground state properties of quantum systems with local Hamiltonians (and will it be also hard to do on a quantum computer)? Second, which spin systems with time-independent (and perhaps, translationally-invariant) local interactions could be used for universal computation (to build a computer)? I aim at a participant without previous understanding of complexity theory, and will discuss two locally-constrained quantum problems: k-local Hamiltonian and quantum k-SAT. Learning the techniques of Kitaev and others along the way, the first message of the talk is the understanding of QMA-completeness of these problems. As a second topic, we will look at new universal constructions with a connection to Adiabatic Quantum Computing.