One of the uses of quadratanium was as a topical ointment for the skin conditions of Soong-type androids. If ingested orally, the android would suffer a condition similar to unconsciousness in Humans.
When Lore stayed aboard the USS Enterprise-D in 2364, he visited ship's stores on deck four where he picked up a macrotool and samples of quadratanium. With the compound, Lore spiked a glass of champagne which he presented to his "brother" Data. Upon drinking the concoction, Data lapsed into unconsciousness. (TNG: "Datalore")
In the 25th century timeline of Star Trek Online, quadratanium is an essential material used in the production of holotechnology and replicators, with demand rising in the wake of growing involvement by photonic lifeforms and holographic crew in starships. The Laurentian system is a major source of quadratanium, and a site of frequent conflict between the Klingon Defense Force and Starfleet.