This might be a very naive question. But what is quantum algebra, really?
Wikipedia defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a satisfying definition. The arXiv admins didn't create a field of mathematics by choosing a name out of nowhere.
Wikipedia (and, in fact, the MathOverflow tag wiki) also lists some subjects: quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory. But again, I don't find this very satisfying, as I feel this doesn't tell me what the overarching idea of quantum algebra is.
(For example, inspired by the table of of contents of the Wikipedia article I could define algebraic topology to be "homotopy, homology, manifolds, knots and complexes". But first, I have certainly missed many subfields of algebraic topology, and second, this is missing the overarching idea, contained in the introduction of the Wikipedia article: algebraic topology is the use of "tools from abstract algebra to study topological spaces". It immediately makes the link behind all the themes I listed clearer, and if I encounter a new theme I can tell if it is AT or not using this criterion.)
This MO question is looking for the intuition behind quantum algebra and relations to quantum mechanics. The main thing I gathered from the answers (that I more-or-less knew already) is that "quantum = classical + ħ", or less informally that we are looking at noncommutative deformations of commutative, classical objects. But this doesn't seem to account for all of quantum algebra. For example, a TQFT is a functor from a category of cobordisms to some algebraic category. Where's ħ? Operadic algebra is also listed as one of the components of quantum algebra, but one can study operads a lot without talking about noncommutative deformations. In fact, I've seen and read many papers about operads listed in math.QA that don't seem to have anything to do with this picture.
In brief: What could be a one-sentence definition of quantum algebra? (In the spirit of the definition of algebraic topology above.)