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We investigate the conditions under which an uncontrollable backgroundprocesses may be harnessed by an agent to perform a task that would otherwisebe impossible within their operational framework. This situation can beunderstood from the perspective of resource theory: rather than harnessing'useful' quantum states to perform tasks, we propose a resource theory ofquantum processes across multiple points in time. Uncontrollable backgroundprocesses fulfil the role of resources, and a new set of objects calledsuperprocesses, corresponding to operationally implementable control of thesystem undergoing the process, constitute the transformations between them.After formally introducing a framework for deriving resource theories ofmulti-time processes, we present a hierarchy of examples induced by restrictingquantum or classical communication within the superprocess - corresponding to aclient-server scenario. The resulting nine resource theories have differentnotions of quantum or classical memory as the determinant of their utility.Furthermore, one of these theories has a strict correspondence betweennon-useful processes and those that are Markovian and, therefore, could be saidto be a true 'quantum resource theory of non-Markovianity'.

Resource theories of multi-time processes: A window into quantum non-Markovianity