The quantum torpedo was a tactical quantum weapon deployed aboard starships that utilized a plasma warhead. Quantum torpedoes were used by several galactic powers including the United Federation of Planets, the Cardassian Union, and in the mirror universe, the Terran Rebellion.
The Federation quantum torpedoes contained a plasma warhead (DS9: "For the Uniform") and used casings similar in shape to photon torpedoes and photonic torpedoes. (DS9: "Valiant") When a ship was destroyed with quantum torpedoes, the explosion left antimatter residue on the hull fragments. (ENT: "Regeneration") Although they were powerful weapons, even a direct hit from a quantum torpedo might not destroy solid neutronium. (DS9: "To the Death")
Starfleet began to include quantum torpedoes within its arsenal as early as 2371. In the same year, the mirror universe Terran Rebellion obtained quantum torpedo technology by stealing it from the Federation, along with the plans to build a Defiant-class starship. (DS9: "Defiant", "Shattered Mirror")
Quantum torpedoes have been carried aboard Defiant- and Sovereign-class vessels, the USS Lakota, and the Federation attack fighters. The Cardassian mobile weapon ATR-4107 also carried quantum torpedoes. The Cardassian torpedoes would have needed modifications in order to be launched from a Federation torpedo launcher. (DS9: "Paradise Lost", "Sacrifice of Angels";
Side note we do not know if the ATR-4107 was upgrade with quatum Torpedoes by Maquis or if Cardassion Union put those types weapons on the Dreadnought missile as part of it design. VOY: "Dreadnought"; Star Trek: First Contact)
In 2371, after hijacking the USS Defiant and taking it deep into Cardassian space, Thomas Riker utilized the ship's quantum torpedoes against a modified Keldon-class cruiser just outside the Orias system. The Defiant fired four torpedoes at the cruiser; two hit the ship's ventral hull, and two hit the tail. The damage from these four torpedoes knocked the ship's defense systems off line and caused the loss of main power. (DS9: "Defiant")
The following year, Captain Sisko used the Defiant's quantum torpedoes against the Klingons, as he attempted to rescue the Detapa Council from the Cardassian ship Prakesh. (DS9: "The Way of the Warrior")
Commander Chakotay noted in 2372 that the missile Dreadnought's quantum torpedoes could be modified to be compatible with the USS Voyager's torpedo launchers, although the weapon was destroyed before the crew could salvage them. (VOY: "Dreadnought")
By mid-year, the USS Lakota had received a major upgrade, including a full complement of quantum torpedoes. Admiral Leyton suggested that Captain Benteen should use the increased firepower provided by the Lakota's quantum torpedoes as a means to destroy the Defiant outright. (DS9: "Paradise Lost")
Later that year, Lieutenant Commander Worf destroyed a Klingon civilian transport ship that suddenly decloaked in front of the Defiant, with a full spread of quantum torpedoes. (DS9: "Rules of Engagement")
In 2373, the USS Enterprise-E used quantum torpedoes in the Battle of Sector 001 to destroy a Borg cube. Four quantum torpedos were used in 2063 over Earth against a Borg sphere that had traveled back in time to sabotage First Contact by destroying the Phoenix. (Star Trek: First Contact)
Later that year, Benjamin Sisko ordered Worf to attach a cargo pod containing two hundred kilograms of trilithium resin onto two quantum torpedoes. These torpedoes were designed to detonate in the atmosphere of Solosos III, causing the planet to be uninhabitable for Humans, and forcing the Maquis colony there to be evacuated. (DS9: "For the Uniform")
In 2375, the Defiant destroyed a Breen warship with a spread of quantum torpedoes during the Second Battle of Chin'toka, before it was destroyed by the Breen's new energy dampening weapon. (DS9: "The Changing Face of Evil")
As the new USS Defiant pushed on towards Cardassia Prime, during the final battle of the Dominion War, Worf reported that the ship had used all but forty-five of its quantum torpedoes. (DS9: "What You Leave Behind")
Background information Edit
Quantum torpedoes were first mentioned and seen in DS9: "Defiant". The effect used to display the torpedoes of the Federation attack fighters attacking the center of the Dominion lines had a yellowish halo instead of the bluish seen elsewhere.
In the first draft of the script of Star Trek: First Contact,  a fleet of Borg ships is defeated with the new quantum torpedoes. Entire Borg cubes are destroyed with single hits. Quantum torpedoes are described as oscillating and changing shape and color as they streak towards their targets, piercing through the hull like armor-piercing shells, exploding the Borg ships from within in an explosion visually resembling a supernova. A similar concept of a highly destructive weapon, the transphasic torpedo, was later on used in VOY: "Endgame".
Technical Manual Edit
The Star Trek: Deep Space Nine Technical Manual states that the basic mechanism of the quantum torpedo was first operated experimentally in 2236. The first warhead application was tested in 2355, afterwards the mass fabrication started. The Federation quantum torpedo model is called "Pho-torp Mark Q-II" and it has a casing differently shaped from the photon torpedo.
The quantum warhead relies on rapid energy extraction from zero-point vacuum. This is established from an 11-dimensional space-time membrane, twisted into a Genus-1 topology string, housed inside the ultraclean vacuum of a 1.38 meter-long teardrop shaped zero-point field reaction chamber. The detonation of a photon torpedo warhead, enriched with fluoronetic vapor, inside the torpedo powers a continuum distortion emitter. It expands the membrane and pinches it out of the background vacuum. The membrane forms into subatomic particles accompanied by a high-explosive energy release.
The statement of quantum warheads and enriched photon warheads seems to contradict the canonical account that quantum torpedoes contain a plasma warhead, from DS9: "For the Uniform". The isoton figures given in the Star Trek: Deep Space Nine Technical Manual are much smaller than the ones stated on-screen, in VOY: "Scorpion, Part II" etc. The enriched photon warhead for example is rated only at 21.8 isotons and the membrane energy potential upon detonation is rated to be only at least 52.3 isotons in the Manual, while basic class-6 photon warheads were rated at two hundred isotons on-screen.
Propulsion system of the quantum torpedo is a warp sustainer engine and four microfusion thrusters. The engine coils of the warp sustainer grab and hold a hand-off warp field from the torpedo launcher tube's sequential field induction coils. A miniature matter/antimatter fuel cell adds power to the hand-off field. When launched in warp flight, torpedo will continue to travel at warp, when launched at sublight, torpedo will travel at a high sublight speed, but will not cross the warp threshold. The quantum torpedo uses a bio-neural gel processor for flight control, and a thoron web to block countermeasure radiation.
Deep Space 9 was armed later on with quantum torpedoes. class-8 and class-9 probe variants also use the quantum torpedo casing. There are apparently also micro quantum torpedoes. (pp. 77, 82, 85, 86, 130 and Star Trek: The Next Generation Technical Manual, p. 129)
The 2002 video game Star Trek: Bridge Commander establishes that quantum torpedoes are available in more limited quantities when compared to the more common photon torpedoes, and as such are used sparingly. A Sovereign-class vessel in the game is depicted wielding a maximum complement of only sixty quantum torpedoes.
The manual for the other 2002 video game Star Trek: Starfleet Command III explains the origins of the torpedo in detail: "The quantum torpedo is an advanced heavy weapon developed by Starfleet as a part of its Advanced Defense Initiative to combat Borg incursions. As the photon torpedo reached what many Starfleet researchers agreed was its highest possible explosive yield, a group of engineers at the Starfleet R&D facility on Groombridge 273-2A began investigating methods for achieving a higher release of energy from a projectile of equal dimensions to the photon torpedo."
In the 2003 video game Star Trek: Elite Force II, the quantum burst (β) launcher is a heavy hand-held anti-infantry weapon, that fires mini-quantum torpedoes. The torpedoes can be guided to their targets using a phased laser sight.
The 2006 video game Star Trek: Legacy also utilizes quantum torpedoes on ships such as Defiant-class and Sovereign-class. The torpedoes can be fired from forward torpedo launchers in spreads of three torpedoes while aft torpedo launchers still use photon torpedoes.
The 2010 MMORPG Star Trek Online features quantum torpedoes. Their characteristic differences from photon torpedoes is higher velocity and higher damage per torpedo, but slower reload rate. Some Federation NPC vessels are equipped with these, as well as all Dominion NPC ships. The assault cruiser refit is outfitted with a wide angle quantum torpedo launcher, which allows players to fire quantum torpedoes at a wider angle and while broadsiding.