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We investigate the evolution of quantum information under Pauli measurementcircuits. We focus on the case of one- and two-dimensional systems, which arerelevant to the recently introduced Floquet topological codes. We define localreversibility in context of measurement circuits, which allows us to treatfinite depth measurement circuits on a similar footing to finite depth unitarycircuits. In contrast to the unitary case, a finite depth locally reversiblemeasurement circuit can implement a translation in one dimension. A locallyreversible measurement circuit in two dimensions may also induce a flow oflogical information along the boundary. We introduce "measurement quantumcellular automata" which unifies these ideas and define an index in onedimension to characterize the flow of logical operators. We find a$\mathbb{Z}_2$ bulk invariant for two-dimensional Floquet topological codeswhich indicates an obstruction to having a trivial boundary. We prove that theHastings-Haah honeycomb code belongs to a class with such obstruction, whichmeans that any boundary must have either nonlocal dynamics, period doubled, oradmits anomalous boundary flow of quantum information.

Measurement Quantum Cellular Automata and Anomalies in Floquet Codes