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 a 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 postprocessing the first instrument into a new one. This is similar to how a measurement described by a positive operator-valued measure (POVM) can be postprocessed into another by way of classical randomization of its outcomes using a stochastic matrix. In this work we study the postprocessing 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 postprocessings between different types of instruments, and draw connections between postprocessings of some of these instruments and their induced POVMs.

What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this leads to difficult static questions about the ground-state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the pinned commuting and pinned stoquastic Local Hamiltonian problems are quantum-Merlin-Arthur–complete. Second, we investigate pinned dynamics and demonstrate that fixing a single qubit via often repeated measurements results in universal quantum computation with commuting Hamiltonians. Finally, we discuss variants of the ground-state connectivity (GSCON) problem in light of pinning, and show that stoquastic GSCON is quantum-classical Merlin-Arthur–complete.