  The large deviations at various levels that are explicit for Markov jump
processes satisfying detailed-balance are revisited in terms of the
supersymmetric quantum Hamiltonian $H$ that can be obtained from the Markov
generator via a similarity transformation. We first focus on the large
deviations at level 2 for the empirical density ${\hat p}(C) $ of the
configurations $C$ seen during a trajectory over the large time-window $[0,T]$
and rewrite the explicit Donsker-Varadhan rate function as the matrix element
$I^{[2]}[{\hat p}(.) ] = \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{{\hat
p}} \rangle $ involving the square-root ket $\vert \sqrt{{\hat p}} \rangle $.
[The analog formula is also discussed for reversible diffusion processes as a
comparison.] We then consider the explicit rate functions at higher levels, in
particular for the joint probability of the empirical density ${\hat p}(C) $
and the empirical local activities ${\hat a}(C,C') $ characterizing the density
of jumps between two configurations $(C,C')$. Finally, the explicit rate
function for the joint probability of the empirical density ${\hat p}(C) $ and
of the empirical total activity ${\hat A} $ that represents the total density
of jumps of a long trajectory is written in terms of the two matrix elements $
\langle \sqrt{ {\hat p} } \vert H \vert \sqrt{{\hat p}} \rangle$ and $\langle
\sqrt{ {\hat p} } \vert H^{off} \vert \sqrt{{\hat p}} \rangle $, where $H^{off}
$ represents the off-diagonal part of the supersymmetric Hamiltonian $H$. This
general formalism is then applied to pure or random spin chains with
single-spin-flip or two-spin-flip transition rates, where the supersymmetric
Hamiltonian $H$ correspond to quantum spin chains with local interactions
involving Pauli matrices of two or three neighboring sites.
