  Given a nonparametric Hidden Markov Model (HMM) with two states, the question
of constructing efficient multiple testing procedures is considered, treating
one of the states as an unknown null hypothesis. A procedure is introduced,
based on nonparametric empirical Bayes ideas, that controls the False Discovery
Rate (FDR) at a user--specified level. Guarantees on power are also provided,
in the form of a control of the true positive rate. One of the key steps in the
construction requires supremum--norm convergence of preliminary estimators of
the emission densities of the HMM. We provide the existence of such estimators,
with convergence at the optimal minimax rate, for the case of a HMM with $J\ge
2$ states, which is of independent interest.
