In 2044, 25-year-old Joe works for a Kansas City crime syndicate as a "looper." Since future technology has made it near-impossible to dispose of bodies, the syndicate uses time travel, invented thirty years later and outlawed instantly. Managed by a future man named Abe Mitchell, loopers kill and dispose of face-concealed victims, and are paid with silver bars strapped to the target. To prevent connections to the syndicate, loopers kill their future selves with gold bars strapped to them when they retire, effectively ending the contract and "closing the loop".
Joe's friend Seth, part of a minority that manifest low-level telekinesis (or TK), confides that his old self has escaped, after warning him of a person in the future called the Rainmaker who will overthrow the five major bosses and close all loops. Joe reluctantly hides Seth in his apartment's floor safe, but is taken to Abe by Kid Blue, one of Abe's elite "Gat Men." Joe reveals Seth's location instead of forfeiting half his silver, and Abe's men cut an address into younger Seth's arm, then begin severing body parts. As Old Seth's limbs disappear, he goes to the address and is killed.
When Joe's next target arrives, it is his older self with his face uncovered. Before Joe can kill him, Old Joe shields himself, knocks younger Joe unconscious and escapes. Returning to his apartment, Young Joe fights with Kid Blue, only to fall off a fire escape and black out.
In another timeline, Young Joe kills his older self as he arrives. He moves to Shanghai, where his drug addiction and partying persist, becoming a hitman to finance himself. Years later, he meets a woman during a bar fight and they marry. Thirty years after, Joe is taken to close the loop and his wife is killed in the process. Overpowering his captors, Joe sends himself back to 2044 thereby altering history. When Old Joe sees Young Joe fall, he shoots the Gat Men and drags him away.
Old Joe begins to manifest vague memories of Young Joe's actions in the present, and meets his younger self at a diner to explain that he intends to save his wife by killing the Rainmaker as a child. Kid Blue and several other Gat Men arrive at the diner and a gunfight ensues: Young Joe collects a corner of Old Joe's map as both escape.
Young Joe follows the map to a farm where Sara and her son Cid live. Sara recognizes the number on the map as Cid's birthday and birth hospital's code. Young Joe guesses that Old Joe is going to kill all three children born at the hospital that day, not knowing which one will become the Rainmaker, so he waits at the farm to protect Cid and Sara.
Jesse, another Gat Man, comes looking for both Joes at the farm, but Cid and Young Joe hide in an underground tunnel. Later that night Sara and Young Joe have sex, and Sara reveals she has TK powers. Cid's powers are revealed to be even stronger, with Sara hiding in a safe when he has a tantrum. In the morning, Young Joe wakes to find Jesse holding Sara at gunpoint in the living room. Frightened, Cid falls down the stairs and telekinetically destroys Jesse. Young Joe realizes that Cid will become the Rainmaker, using his powers to control the city, and that Old Joe will now know this from his memories.
Kid Blue captures Old Joe and takes him to Abe. Old Joe breaks free and kills Abe and his henchmen, then travels to Sara's farm. While Young Joe kills Kid Blue, Old Joe pursues Sara and Cid. Cid's cheek is grazed by Old Joe's bullet, and he creates a telekinetic blast, but is calmed by Sara before he can kill them. Telling Cid to run into the cane field, Sara stands between Old Joe and her son. Young Joe realizes that Cid's mother's death will turn him into the Rainmaker and commits suicide, erasing Old Joe's existence, saving Sara and preventing Cid from becoming the Rainmaker.

Question: When was the time travel that the syndicate uses invented?
Answer: 30 years before
Alternative answer: In 2074.