The quantum rod was a key piece of technology in the drive system of the SS Trojan vessel.
When the crew of Red Dwarf encountered the SS Trojan, Rimmer was impressed by the fabulous technology used in its construction. He was especially happy to have a "Greeny-glowy thing", which he found on the bridge by the captain's chair. Kryten identified it as the ship's quantum rod.
The principles of its operation were complex, but Kryten was able to explain them in layman's terms. The rod worked like a magnet, drawing back together matter that had once been connected. Since all matter had been connected together before the Big Bang, the rod could detect seemingly distant matter and jump from place to place by contracting space-time. It was a delicate piece of hardware and Kryten insisted that Rimmer replace it carefully.
The rod's ability to reconnect previously distant things may have been what caused Rimmer's brother Howard to make contact with the SS Trojan moments later. The odds of two Rimmer brothers reuniting after three million years and astronomical distances were astoundingly small.
After it was revealed that Sim Crawford had killed the crew of the Columbus 3, she explained that her new plan was to use the quantum rod's abilities to unite all simulants and conquer the universe. The Cat was able to stop her by overloading her core program with Howard's resentment file. ("Trojan")
Aboard Red Dwarf
Later, Kryten was experimenting with crystals from the quantum rod aboard Red Dwarf. This caused he and The Cat to become quantum entangled, enhancing their ability to synchronise during times of stress. The crystals invigorated their PSI, as posited by Arthur Koestler in his book "The Roots Of Coincidence".
Kryten and The Cat began talking at the same time and were more aware of coincidences, such as the fact that the BEGGs would choke to death. They used their powers of synchronicity to find the co-ordinates to the ERRA station. The effects of the rod's crystals wore off eventually. ("Entangled")