[creesch]

> I want to know what future technologies could accelerate an object to a significant fraction of the speed of light.

With all due respect, this has been extensively debated online and is extremely easy to look up.

> Could those breakthroughs be achieved in the next 2 or 3 centuries?

Any predictions on future scientific breakthroughs are wishful thinking. Even directions that seem promising at some point can end up being dead ends and we simply don't know what we will or will not achieve.

With current technologies, even hypothetical ones we can't get there in a time period that is under a few centuries at best and over a millennium at worse. Even for the centuries figure we'd require technologies we haven't been able to crack yet.