Asklemmy

You can transmit something, but it has a noise added to it. To decode it, you need to send the readings to the other end, via normal means. Basically, the receiver can tell, in hindsight, that a message was sent, but only once its other half has been received via normal means. The best you can do is get a timestamp of when the message was sent, as well as a message channel that is impossible to intercept.

The problem comes when QM meets relativity. With instant communication, you can send information into its own past. E.g. A and B are 2 planets. C is a ship, passing planet B at relativistic speeds. Planet A sends a message to B, over the FTL link. B then sends it to C, over a normal link. C, finally sends it back to A over FTL. Due to the 'tilt' of C's light cone, the "now" of A-C is behind the "now" of A-B. This allows for paradoxical situations. The maths of Relativity implies that you can't form a closed time loop like this. Such behaviours tend to imply some deeper rule, even if we haven't found its cause yet.

Quantum mechanics has a lot of strangeness. It also seems to play fast, but not loose with causality. E.g. objects can move backwards in time, but still obey causality. Others can be smeared over time space, but still collapse to a causality obeying state. Etc

Quantum theory is only "controversial" to the general public, mainly because we haven't found a way to explain in simple terms things like superposition, entanglement, quantum tunneling. Quantum theory is spectacularly successful, though incomplete.

Even the "simple" stuff like the uncertainty principle takes a detailed understanding to properly grasp why there are pairs of properties that are inherently linked, and that information about one dictates how much you can know about the other. e.g. position/momentum and energy/time.