The quantum torpedo was originally conceived as a replacement for the standard photon torpedo. Development began in the late 2360s, and started to enter use in the early 2370s. (DS9 episode: "Defiant", ST movie: First Contact)
The quantum torpedo utilizes a zero-point energy reaction to create a 50+ isoton explosion. The basic mechanism was already being experimented with in the year 2236. The first warhead application came in 2355. It was created as one of the weapon systems designed to fight the Borg Collective. Indeed, quantum torpedoes were widely used by the Federation fleet when the Borg launched their second assault on Earth in early 2373, one of the starships that used these weapons was the USS Enterprise-E. There are two types of torpedo tubes created for this weapon: The pulse fire tube (1 torp/second) and the rapid fire tube (which can fire at a rate of 4/second) (DS9 reference: Technical Manual; ST movie: First Contact)
After their successful first test on the USS Defiant, quantum torpedoes were implemented throughout the Federation fleet. Defiant, Sovereign, Luna, Vesta, Akira and Achilles-class starships included quantum torpedoes in their arsenal. The technology was later shared with the Klingon Empire and was implemented on their Qang-class vessels.
In the Kelvin timeline, Starfleet had begun using quantum torpedoes as early as the 2250s. Like in the prime reality, the basic mechanism was tested in 2236. Obviously, the Federation developed the quantum torpedo early because the Romulans had acquired the Narada's schematics as a result of a Vulcan revenge plot gone wrong and used them to create their own ships. While not yet in use as standard armaments they could be deployed on Arbiter-class, Constitution-class, Defender-class, Guardian-class, and Interceptor-class starships. (TOS comic: "Vulcan's Vengeance, Part 2", TOS video game: D-A-C)
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