In this talk, I will present a novel shape-constrained estimator of the autocovariance sequence resulting from a reversible Markov chain.  A motivating application for studying this problem is the estimation of the asymptotic variance in central limit theorems for Markov chains. Asymptotic variance is a key quantity in quantifying the uncertainty of the sample mean from Markov chain iterates, so accurate estimation of asymptotic variance has both statistical and practical significance. Our approach is based on the key observation that the representability of the autocovariance sequence as a moment sequence imposes certain shape constraints, which we can exploit in the estimation procedure. I will discuss the theoretical properties of the proposed estimator and provide strong consistency guarantees for the proposed estimator. Finally, I will empirically demonstrate the effectiveness of our estimator in comparison with other current state-of-the-art methods for Markov chain Monte Carlo variance estimation, including batch means, spectral variance estimators, and the initial convex sequence estimator.