  Studying sequential measurements is of the utmost importance to both the
foundational aspects of quantum theory and the practical implementations of
quantum technologies, with both of these applications being abstractly
described by the concatenation of quantum instruments into a sequence of
certain length. In general, the choice of instrument at any given step in the
sequence can be conditionally chosen based on the classical results of all
preceding instruments. For two instruments in a sequence we consider the
conditional second instrument as an effective way of post-processing the first
instrument into a new one. This is similar to how a measurement described by a
positive operator-valued measure (POVM) can be post-processed into another by
way of classical randomization of its outcomes using a stochastic matrix. In
this work we study the post-processing relation of instruments and the partial
order it induces on their equivalence classes. We characterize the greatest and
the least element of this order, give examples of post-processings between
different types of instruments and draw connections between post-processings of
some of these instruments and their induced POVMs.
