Estimating the distribution of the hitting time to a rarely visited set of states presents substantial challenges. We recently designed simulation-based estimators to exploit existing theory for regenerative systems that a scaled geometric sum of independent and identically distributed random variables weakly converges to an exponential random variable as the geometric's parameter vanishes. The resulting approximation then reduces the estimation of the distribution to estimating just the mean of the limiting exponential variable. The present work examines how randomized quasi-Monte Carlo (RQMC) techniques can help to reduce the variance of the estimators. Estimating hitting-time properties entails simulating a stochastic (here Markov) process, for which the so-called array-RQMC method is suited. After describing its application, we illustrate numerically the gain on a standard rare-event problem. This chapter combines ideas from several areas in which Pierre L'Ecuyer has made fundamental theoretical and methodological contributions: randomized quasi-Monte Carlo methods, rare-event simulation, and distribution estimation.