The central topics of this thesis are operating characteristics for binary hypothesis testing in classical and quantum settings and overcomplete quantum measurements for quantum binary state discrimination. With this we explore decision and measurement operating characteristics defined as the tradeoff between probability of detection and probability of false alarm as parameters are varied. The thesis specifically addresses the Neyman-Pearson optimality of receiver operating characteristics when they are generated using threshold tests on the score variable rather than threshold tests on the likelihood ratio. The analysis applies to any scalar score variable. In the quantum setting, informationally overcomplete POVMs are explored to provide more robust quantum binary state discrimination schemes. We focus on equal trace rank one or Etro POVMs, which can be specified by arrangements of points on a sphere that we refer to as an Etro sphere.Ph.D.Sc.D.