The temporal weapon was a ship-mounted directed energy weapon using temporal science as a basis for the weapon. Being in a state of constant temporal flux, standard starship deflector shields proved ineffective against the temporal weapon.
At an unknown time, Doctor Samantha Delaney returned to the prime reality having stolen the Dark Starship from an alternate reality. She outfitted the Dark Starship with a temporal weapon, which she began testing on larger and larger subjects.
While in 2360, the temporal weapon threw the USS Cartwright (NCC-42953) forward in time. The battle damaged the Cartwright so severely that the saucer section had to separate from the stardrive section and crash land on the planet's surface. (Star Trek: The Cantabrian Expeditions: "Bitter Sweet")
In 2373, the temporal weapon struck the port forward cockpit of a runabout, swapping the prime reality version of Elizabeth Singh with her Stranger Than Fiction reality counterpart. (Star Trek: The Cantabrian Expeditions: "Stranger Than Fiction")
Later that same year, the USS Aotearoa (NX-78312) bore the full brunt of a blast from the temporal weapon intended for the USS Cantabrian (NCC-24705). The blast sent the Aotearoa into an unknown time and space. (Star Trek: The Cantabrian Expeditions: "Best Served Cold") The remnants of the discharge allowed the Cantabrian crew to communicate with the Aotearoa crew until the remnants finally collapsed. (Star Trek: The Cantabrian Expeditions: "I Who Have Nothing")
Delaney wanted to test the temporal weapon on a larger target, so she fired it on the New Yoff colony, destroying the colony and killing all the inhabitants, including Cantabrian chief medical officer Marie Bourget's parents, Michel and Dominique Ndoye. (Star Trek: The Cantabrian Expeditions: "Homeland")
With the test on a planet successful, Delaney steered the Dark Starship towards Earth to fire the temporal weapon there. Many Starfleet vessels fell to the Dark Starship, including the USS Laramie (NCC-73908) under Captain Anne Lansing's command, but the Cantabrian crew were able to stop the temporal weapon when Cantabrian captain Noah Wrightson threw himself into the stream, overloading the weapon in the process. (Star Trek: The Cantabrian Expeditions: "By All Means Necessary")
In reality, Wrightson was thrown back in time to ancient Egypt. (Star Trek: The Cantabrian Expeditions: "Where There's a Man Who Has No Voice, There Ours Shall Go Singing")