We derive new concentration bounds for time averages of measurement outcomes
    in quantum Markov processes. This generalizes well-known bounds for classical
    Markov chains which provide constraints on finite time fluctuations of
    time-additive quantities around their averages. We employ spectral,
    perturbation and martingale techniques, together with noncommutative $L_2$
    theory, to derive: (i) a Bernstein-type concentration bound for time averages
    of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type
    concentration bound for the same process, (iii) a generalization of the
    Bernstein-type concentration bound for counting processes of continuous time
    quantum Markov processes, (iv) new concentration bounds for empirical fluxes of
    classical Markov chains which broaden the range of applicability of the
    corresponding classical bounds beyond empirical averages. We also suggest
    potential application of our results to parameter estimation and consider
    extensions to reducible quantum channels, multi-time statistics and
    time-dependent measurements, and comment on the connection to so-called
    thermodynamic uncertainty relations.

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