  Currently, there is no systematic way to describe a quantum process with
memory solely in terms of experimentally accessible quantities. However, recent
technological advances mean we have control over systems at scales where memory
effects are non-negligible. The lack of such an operational description has
hindered advances in understanding physical, chemical and biological processes,
where often unjustified theoretical assumptions are made to render a dynamical
description tractable. This has led to theories plagued with unphysical results
and no consensus on what a quantum Markov (memoryless) process is. Here, we
develop a universal framework to characterise arbitrary non-Markovian quantum
processes. We show how a multi-time non-Markovian process can be reconstructed
experimentally, and that it has a natural representation as a many body quantum
state, where temporal correlations are mapped to spatial ones. Moreover, this
state is expected to have an efficient matrix product operator form in many
cases. Our framework constitutes a systematic tool for the effective
description of memory-bearing open-system evolutions.
