Abstract Semantic automata theory studies the complexity of generalized quantifiers in terms of the string languages that describe their truth conditions. An important point has gone unnoticed so far: for most quantifiers that are determiners, these string languages are subregular. Whereas quantifier phrases such as an even number of correspond to specific regular languages, every, no, some and not all stay within the much weaker class of tier-based strictly local languages (TSL). In addition, it seems to hold crosslinguistically that a TSL quantifier may be expressed by a single lexical item only if its tier projection is monotonic. This constitutes a novel typological universal that limits how complex a meaning may be stored in a single lexical item. TSL is also known to play a central role in phonology, morphology, and syntax, which suggests that subregularity in general and tier-based strict locality in particular may be a computational universal of natural language that surfaces across all its submodules, including semantics.