We study persistence probabilities of Hermite processes. As a tool, we derive a general decorrelation inequality for the Rosenblatt process, which is reminiscent of Slepian's lemma for Gaussian processes or the FKG inequality and which may be of independent interest. This allows to compute the persistence exponent for the Rosenblatt process. For general Hermite processes, we derive upper and lower bounds for the persistence probabilites with the conjectured persistence exponent, but with non-matching boundaries.