Criteria for generalized macroscopic and mesoscopic quantum coherence

We consider macroscopic, mesoscopic, and "S-scopic" quantum superpositions of eigenstates of an observable and develop some signatures for their existence. We define the extent, or size S of a superposition, with respect to an observable x, as being the range of outcomes of x predicted by that superposition. Such superpositions are referred to as generalized S-scopic superpositions to distinguish them from the extreme superpositions that superpose only the two states that have a difference S in their prediction for the observable. We also consider generalized S-scopic superpositions of coherent states. We explore the constraints that are placed on the statistics if we suppose a system to be described by mixtures of superpositions that are restricted in size. In this way we arrive at experimental criteria that are sufficient to deduce the existence of a generalized S-scopic superposition. The signatures developed are useful where one is able to demonstrate a degree of squeezing. We also discuss how the signatures enable a new type of Einstein-Podolsky-Rosen gedanken experiment.