  Consecutive measurements performed on the same quantum system can reveal
fundamental insights into quantum theory's causal structure, and probe
different aspects of the quantum measurement problem. According to the
Copenhagen interpretation, measurements affect the quantum system in such a way
that the quantum superposition collapses after the measurement, erasing any
knowledge of the prior state. We show here that counter to this view,
unamplified measurements (measurements where all variables comprising a pointer
are controllable) have coherent ancilla density matrices that encode the memory
of the entire set of quantum measurements, and that the quantum chain of a set
of consecutive unamplified measurements is non-Markovian. In contrast,
sequences of amplified measurements (measurements where at least one pointer
variable has been lost) are equivalent to a quantum Markov chain. An analysis
of arbitrary non-Markovian quantum chains of measurements reveals that all of
the information necessary to reconstruct the chain is encoded on its boundary
(the state preparation and the final measurement), reminiscent of the
holographic principle.
