  The term "Layers of classicality" in the context of quantum measurements, was
introduced in [T. Heinosaari, Phys. Rev. A 93, 042118 (2016)]. The strongest
layer among these consists of the sets of observables that can be broadcast and
the weakest layer consists of the sets of compatible observables. There are
several other layers in between those two layers. In this work, we study their
physical and geometric properties and show the differences and similarities
among the layers in these properties. In particular we show that: (i) none of
the layers of classicality respect transitivity property, (ii) the concept like
degree of broadcasting similar to degree of compatibility does not exist, (iii)
there exist informationally incomplete POVMs that are not individually
broadcastable, (iv) a set of broadcasting channels can be obtained through
concatenation of broadcasting and non-disturbing channels, (v) unlike
compatibility, other layers of classicality are not convex, in general.
Finally, we discuss the relations among these layers. More specifically, we
show that specific type of concatenation relations among broadcasting channels
decide the layer in which a pair of observables resides.
