A Short Conspiratorial History of Time Travel
It’s hard to imagine a technology that could be more disruptive to every facet of life on earth than time travel. The ability to step back in time and change something, anything, however noble your intent is inherently dangerous. It is inherently dangerous not just to you, but to the very fabric of reality. Indeed, should such technology exist it would have to be used with utmost care, with a careful understanding of the rippling consequence of each and every action and reaction in history. That’s why they tell you it’s impossible. That’s why they’re hiding it from you.
19th and 20th Century Time Machines
The earliest known time machine was discovered (invented?) quite accidentally by famed inventor Nikola Tesla in March 1895, when he was struck by a 3.5 million volt resonant vorticular electromagnetic current (generating a highly charged rotating electromagnetic field) forcing him out of his normal space-time frequency. From his temporary vantage point outside the timeline, Tesla was able to see the immediate past, present, and future with perfect clarity, and evidently also came to understand that a longer stay outside time would wreak permanent damage upon his mind and body. His Fifth Avenue lab burned down shortly thereafter, taking with it dozens of prototypes, designs, as well as notes and research that Tesla had gathered since coming to New York. The fire was, of course, ruled an accident. But was it?
It is unclear how much Tesla continued to experiment with time-displacement (a necessary precursor for time-travel) in the years after the Fifth Avenue fire, as his notes and personal effects were confiscated for further study by the U.S. government upon his death in 1943. The probably sanitized versions of Tesla's notes and journals the government eventually released to the public made no further mention of Tesla's time-displacement experiments, but that does not mean that such notes do not exist. In fact, powerful evidence exists to the contrary.
The Philadelphia Experiment
By the fall of that year, a group of scientists in the employ of the U.S. military -- including, allegedly, Albert Einstein -- had taken Tesla's work and expanded it form the basis of Project Rainbow, an effort at introducing stealth technology to WWII in the form of actual invisibility. The idea was that Tesla's rotating vorticular electromagnetic currents could be used to charge the hull of a naval vessel, temporarily changing it's space-time resonant frequency in such a way that the ship would be rendered invisible to enemy mines, torpedoes, and even to the naked eye.
Things did not go entirely according to plan. When the hull was charged, the ship -- typically, the USS Eldridge or sometimes USS Eldritch -- disappeared from view. However, it reappeared instantly several hundred miles to the south, just outside the shipyards at Norfolk, Virginia. It was in Norfolk that the Eldridge was sighted by the SS Andrew Furseth, a surprised civilian vessel that quickly moved in closer to investigate. The Andrew Furseth's crew's report were stranger yet: some men appeared to be fused with the bulkheads on the upper decks of the Eldridge; other men were apparently on fire, burning (and screaming) but not consumed.
Then, as suddenly as it appeared, the Eldridge once more blinked out of existence, only to return to Philadelphia a full ten seconds earlier than the recorded time when it left Norfolk. Stealth may have been problematic, but time travel had been again achieved. It's probably worth noting that the historicity of Project Rainbow is regularly (and vigorously) denied by the U.S. Naval establishment, but they would not protest if they had nothing to hide.
A Move to Montauk
Although the Philadelphia experiment had failed to produce a usable stealth prototype, it had made important steps in operationalizing time displacement technology. Those steps required further development: Project Rainbow would continue, funded by captured Nazi gold and hidden within New York's Brookhaven National Laboratories for the next few years. Under their auspices, Project Rainbow expanded to include not only time travel research, but also mind control programs, psychic power development, remote viewer training, reverse-engineering captured UFOs, and according to some sources, the origins of the U.S. government's Men in Black program.
By the late 1960s, the program had grown too large for Brookhaven't facilities; the Project Rainbow's personnel and equipment were moved to an underground facility beneath a soon-to-be-decommissioned Air Force Base in Montauk, New York, where it resumed operations as Project Phoenix. By the mid-1980s, Project Phoenix had evolved, giving way not only to more consistently successful time travel experiments, but also quantum teleportation research that ultimately allowed some program subjects to teleport themselves through both space and time. It was at this time, allegedly, that the program -- now called Project Pegasus -- began to explore and colonize a far-future (or possibly distant past) Mars.
Game Time
Time travel makes a particularly useful theme for a conspiracy-themed adventure because its very potential existence leaves your players unable to completely trust any facts they are presented with. A key inventor -- like Tesla, in our example above -- could have fallen victim to the machinations of jealous or greedy contemporaries. Or maybe the 5th Avenue fire was engineered much later, by a secret cabal of future time travelers to eliminate a potential rival, or even just to ensure that the volatile technology would remain in their own control.
Even in the "real life" histories of these secret programs, conflicting accounts, overlapping timelines, and faked deaths abound. Some accounts have Tesla himself leading Project Rainbow well into the 1990s, while others place famous physicist John Von Neumann in charge of the program (apparently using his time travel technology to lead a parallel but separate public life as a reputable scientist and rocketry expert). You can set a game inside these secret programs -- as agents being brought up through the ranks to investigate temporal anomalies, or even to fight a Martian border war alongside a future President.
But at the end of the day, they can't trust anyone. How can you, when you can't even be sure that the past you know is really the past?