  Building on earlier work, we further develop a formalism based on the
mathematical theory of frames that defines a set of possible phase-space or
quasi-probability representations of finite-dimensional quantum systems. We
prove that an alternate approach to defining a set of quasi-probability
representations, based on a more natural generalization of a classical
representation, is equivalent to our earlier approach based on frames, and
therefore is also subject to our no-go theorem for a non-negative
representation. Furthermore, we clarify the relationship between the
contextuality of quantum theory and the necessity of negativity in
quasi-probability representations and discuss their relevance as criteria for
non-classicality. We also provide a comprehensive overview of known
quasi-probability representations and their expression within the frame
formalism.
