This almost seems too basic though. Once at the levels of being able to do particle entanglement, aren't you kind of expected to have stuff like lasers and solid-state electronics? You could use something akin to frequency shift keying, but instead of two frequencies it's angular shifting of circular polarization on the laser acting upon the entangled particles in the excited medium. (FSK is a protocol that exists because you want to differentiate the 0's in binary from the "dead air" of non-transmission. More or less a rudimentary form of error checking.) You could start out at a rather low baud rate to establish that an artificial signal is being produced and perhaps even get as far as a handshake protocol, but from there you could ramp it up all the way to Gb/s that's normal with a commercial fiber-optic modem. No idea of what latency would be, but in theory one could push a whole encyclopedia of stuff through there in a few days given the nature of an optical carrier like that. In which case the messages can be as detail rich as you're willing to make it. (Still some issues of encoding, not like everyone in the galaxy would know what to make of a .PNG, .MKV, or .MP3 - but eventually it could get worked out.)

Or am I dumb for thinking that? (Trying to picture how to come at it from technological approach.) I mean we have the stuff on the shelves out there to build a proper modem, it wouldn't have to resort to stuff that goes back to the keyed telegraph days. (Although from some perspectives here, that may put some extraterrestrials in disbelief wondering why we're even stuck on this backward seeming rock. So perhaps making it overly basic is intentional to get some semblance of help?)

Might want to look at free-electron lasers and their ability to excite THz wavelengths. I think that may be able to produce pairing in two separate regions of plasma at the excited state. (Now consider pairing plasma in one container you keep locally, then perhaps trying for the nuclear reactor in the sky that nature provides on the other end.)

That's a bit beyond me though other than speculation, since this topic made me only curious enough to scratch the surface. This seems like a possible subject if you were to dig into a rabbit-hole if looking into a particular and relatively new science using the search term "quantum metrology". Then if you go down the avenues that deal with entanglement, perhaps that may lead to something?

If you're just trying to listen in only it might be more accessible. (Although it'd still take a rather high allowance hobby, yet there are people on YouTube that make videos with industrial CNC machines - so it may not be out of reach for the odd hobbyist out there.) The "crystals" may be off-the shelf optical components used with lasers like Wolloston prisms and such. Then it's down to looking for things like interference patterns and using optical sensors to read them. More along the lines of building your own telescopes, but then hooking up custom built analysis tools with off-the-shelf parts.

Looks like we have the technology (as far as humanity goes), but it's just a matter of putting it all together in a way that's not readily intuitive if you don't understand a particular application they could be applied towards. Look at the stuff that Hertz, Ampere, Faraday, and Maxwell were playing with. But we didn't get radio until the likes of Tesla and Marconi also started playing with that stuff.

Just a casual wanderer following links to see what this is about...

Haphazard guess?

You can take something like light from a star, run it through a prism and get a spectrograph right? Also you can run light through polarized filters so you can get "left handed" or "right handed" beams of light, no? So what would happen if you ran two different filters on such light to produce separation, then did a spectrograph on each, and then did something to observe the interference patterns produced between them? If the spectrograph lines manipulated in that manner seem to be dancing in some unusal way, then it might be something to try and run through digital signal analysis and see if there's anything to it. (Occasional runs of counting in binary would be one of the most obvious hints of there being an artificial signal being carried across a particular band. Although there may be some other methods not quite as idiot-proof.)

Of course there may be nothing there and such a process would be chasing a loose end. But it could be a possible way to listen, even if you can't make immediate sense of any encoding process used.

Sending is another matter entirely. But the idea of crystals and entangled photons may hint in a particular direction of what to look for and take advantage of some optical tools that are relatively common already. (At least if you accept there's more to this than some strange cooked up fiction because people are bored and human nature is crazy like that.)

The resilience, the spin (via meditation within?), oily skin, and not minding one's own hairiness even looking a bit wild on the pits and the chin.

Just an amused wanderer when it comes to these parts, but might be somewhat there already if any of those things count.