Higher Teichmüller theories are spaces of representations of surface groups which are discrete, faithful, and stable under deformations. Spaces of positive representations are examples of such theories and have the additional property that they are closed, so they form connected components. One important tool in studying these representations is the notion of a positive flag curve, a special type of curve in the flag manifold G/B. Specializing to the Lie group Sp(4,R), I will describe a discrete analog of positive curves and then explain how all such curves are obtained by lifting a simple closed curve in the 2-sphere with monotone curvature function. This is joint work with Ryan Kirk.