AbstractConsider an ‐vertex graph where each vertex holds one of two distinct opinions. We are interested in the consensus time of synchronous voting processes, where each vertex is allowed to update its opinion according to a predefined commonly local updating rule. This article proposes a general class of voting processes called quasi‐majority functional voting and gives upper and lower bounds of the consensus time. Our results cover many previous results on specific processes, for example, best‐of‐two (a.k.a. 2Choices) and best‐of‐three (a.k.a. 3Majority), as special cases. Our key ingredient is a nonlinear extension of the expander mixing lemma, which enables us to estimate expected and variance changes in opinions of quasi‐majority functional voting on expander graphs.