An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. Even in cases where the convergence time is known to be polynomial, the theoretical bounds are often too crude to be practical. Thus, practitioners like to carry out some form of statistical analysis in order to assess convergence. This has led to the development of a number of methods known as convergence diagnostics which attempt to diagnose whether the Markov chain is far from stationarity. We study the problem of testing convergence in a number of settings and prove that the problem is hard in a computational sense. This is joint work with Andrej Bogdanov and Elchanan Mossel.