  Majorization is a fundamental model of uncertainty with several applications
in areas ranging from thermodynamics to entanglement theory, and constitutes
one of the pillars of the resource-theoretic approach to physics. Here, we
improve on its relation to measurement apparatuses. In particular, after
discussing what the proper notion of second law in this scenario is, we show
that, for a sufficiently large state space, any family of entropy-like
functions constituting a second law must be countably infinite. Moreover, we
provide an analogous result for a variation of majorization known as
thermo-majorization which, in fact, does not require any constraint on the
state space provided the equilibrium distribution is not uniform. Lastly, we
discuss the applicability of our results to molecular diffusion and catalytic
majorization. In this regard, we consider a variation of majorization used in
plasma physics as a model of molecular diffusion and show that no finite family
of entropy-like functions constituting a second law of molecular diffusion
exists. Moreover, we show how our results are useful when dealing with a
conjecture regarding catalytic majorization (i.e. trumping). In particular, we
show that the sort of characterizations of trumping that have been considered
before require an infinite family of real-valued functions.
