A quadister was a component of a Federation starship's auxiliary power console. It was nearly microscopic in size.

In 2273, Captain Will Decker discovered a quadister had burned out, which kept the USS Enterprise's transporters from functioning. Decker wondered why this particular burn-out had occurred, and why it hadn't shown up in the tests, but due to the V'Ger crisis there was no time for further overload tests. Soon afterwards, Sonak and Lori Ciana were killed in a transporter malfunction. (TOS novelization: The Motion Picture)

According to Ex Machina, the faulty module which had taken the transporters offline was a simple backup sensor for error-checking the Heisenberg compensator.
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