Quads are mentioned often in the Star Trek universe, in varying orders of magnitude (kiloquad, megaquad, gigaquad, etc.).
What exactly is a quad, as it relates to modern technology?
(The Memory Alpha article seems limited.)
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Two quotes from the link you've provided answer this. First, it is not a consistently used amount:
The terms quads and kiloquads in TNG were used in a manner consistent with the system defined in the Technical Manual. However, by the time Voyager was airing, they started using extremely large numbers that lacked internal consistency, such as "billions of gigaquads" and "billions of teraquads."
Secondly, it is a deliberately vague amount, since:
The terminology "quad" was used to detract from comparisons possible with modern-day computing power, since reality frequently outstrips fiction when it comes to computer science.
If my memory serves me correctly, this second quote was also referenced in the TNG technical guide.
My understanding of a "quad" is a "quantum digit", equivalent to what is known in real-world computer science as a "qubit".
Like a classical electronic "bit" (binary digit), a qubit has two possible states; on and off. However, unlike a classical bit, a qubit can be both on and off at the same time; technically, it's at some state between them, and the set of all possible states is described by the surface of a unit sphere in 3-dimensional space. This is analogous to quantum theory, with Young's Experiment and the famous "Schrodinger's Cat" analogy. This allows a qubit to store vastly more state information, and also allows for algorithmic processing in which a bit can be considered to be both on and off at the same time, necessary for applications like probability mechanics, large number theory, and cryptography.
As of the writing of all of the TNG-era series (TNG, DS9, Voyager), this branch of comp sci was pretty well-defined if not yet fully realized, and was probably drawn from to conceptualize the computers aboard a starship. As of right now you can count the number of qubits contained in our most advanced quantum computers on the fingers of both hands; by the 24th century, given the exponential pace of technological development, they're dealing with data in the billions (giga), trillions (tera) and even quadrillions (peta) of "quads".
it seems that the terms are incorrect for describing 3d "crystalline" matrix computing. now, IIRC, it was referenced as units called "quads" in which data was manipulated. this unit was a 3d construct of a 4x4x4 "bit" arranged in a 4x4x4 "byte" called a "quad." which would make it a 262144 bit unit.