In logic circuit diagrams, I've seen various conventions for naming inputs and outputs of logic gates and combinatorial circuits. However, stateful elements like latches and flip-flops often have their "state" called Q. I suspect there is a connection with abstract Finite-State Machines from theoretical computer science, where "state" is often noted Q as well (so I asked them too :-)
But why have people picked this particular letter ?
Answer
Alan Turing used the letter q to denote states in what came to be known as Turing machines. Presumably the q stood for quanta, emphasizing a state's discrete rather than continuous nature. This happened in the 30s when quantum theory was permeating the scientific æther.
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