An arms race was a competition between opposing groups; both competing to finish development of military technology before the other.
The Earth Cold War, between the United States and the Soviet Union during the latter half of the 20th century, led to the development of increasingly powerful nuclear weapons. (ENT: "Proving Ground", "Carbon Creek";TOS: "Assignment: Earth"; DS9: "Little Green Men")
In 1968, Gary Seven intended to detonate the nuclear warhead of a United States orbital nuclear warhead platform at least one-hundred miles above the surface of planet Earth to frighten them out of the arms race. Detonating at 104 miles above ground, it resulted in a new and stronger international agreement against the use of such weapons. (TOS: "Assignment: Earth")
A temporal arms race between Section 31 and the Klingon Empire during the 2230s culminated in the development of the Red Angel suit and the Klingon raid on Doctari Alpha in 2236. (DIS: "Project Daedalus", "The Red Angel")