The quantum marginal problem asks whether a family of quantum marginals are compatible with a global quantum state. It is of central importance to a wide range of topics in both quantum many body physics and quantum information. Often it can be the case that when a family of quantum marginals are compatible with a global quantum state, that global state is unique. This thesis explores the role of uniqueness in the quantum marginal problem, in particular the distinction between uniqueness among pure states and uniqueness among all states. A particularly important vari- ant of the quantum marginal problem is the symmetric extendibility problem. This thesis makes headway on characterizing the extremal points of a particular class of symmetric extendibility problems, and demonstrates the importance of understand- ing the distinction between being uniquely determined among mixed states and being uniquely determined among pure states in making further progress in this direction.