Quantum Fugue

If you have a couple of telepaths and the right equipment, you too can send messages over insane distances as long as you're willing to take over the lives of a random person for the day. And I actually don't mean by boosting the telepathic signal to catch someone near the location of where you mean and then take them over a bit so you can have a conversation with the person you need to talk to.
The technology is convoluted, to put it mildly, but it is the most reliable way to get a message across the galaxy in short order. You grab your telepath, and I take this in the loosest form imaginable. Even someone with simple empathic sensitivities is enough to be able to activate it, which is part of what makes is as effective as it is. The technology boosts their abilities, then piggybacks on the nearest suitable candidate.
There is a particular mental state that the receiver has to be in. It's rather arbitrary on the surface, and takes into account certain mental structures and also emotional state. You cannot completely prevent yourself from being a receiver.
The receiver will then go into the quantum dimension (which has shockingly little to do with what science has termed the quantum state) and within this state will then go on a journey to connect to the electronic receiver at the other end and deliver the message. Apparently within the state it can take quite a while, even if it appears to be nearly instantaneous on the outside.
Once the message is delivered, the receiver is thrown back to where they were with no memory of what just happened. (Unless they have some telepathic potential in which case some memories are retained which is how we know something about the process).
Occasionally people have been known to get stuck there, and the factors which cause this have yet to be properly determined. But still, it's really fast and reliable compared to literally any other type of communication at that distance. Fortunately for the average person, most people don't need to send a message across the galaxy in a hurry.