If cryptographic hashes are completely unique, could they theoretically be used to transfer data? I know it sounds ridiculous, but hear me out. We know that good hashing algorithms would generate hashes that are pretty much unique. If, in the future, we create a perfect (or just really near perfect) algorithm, could those hashes be used to represent larger chunks of data in a totally unique way? If they truly were unique and determined entirely by the file, could a powerful enough machine work backwards and reconstruct the file?

Now, this is a bit of a stretch from what we currently know, but I'd think it might be possible. Quantum computers can perform many different tasks at once, and we know that they pose a significant threat to modern forms of encryption because of this ability. While that isn't exactly the same thing as guessing every possible file that could've produced a particular hash, but I'd imagine there might be a way to make the process more efficient.

Maybe you get a hash that's a couple kilobytes long, allowing for billions of different input types. Combine that with a computer capable of generating and testing every possible input simultaneously and you'd be able to send tons of information in a couple kilobytes of transmission.

I'm not asking if it's possible to do with our current technology, as I'm fairly sure it isn't. I just want to know if the process of hashing a file could (in theory) support it.
No.

They can be used to identify data you've stored elsewhere.  For example, you can probably find a preimage under MD5 for 6f5902ac237024bdd0c176cb93063dc4 by searching the web, and it might even be meaningful English text.  And you probably won't find more than one preimage.

This is the idea of content-addressed storage, which is widely used, e.g. by Tarsnap—although it would be a mistake to use MD5 or SHA-1 for content-addressed storage, because collisions are readily available to anyone who wants them, which have hilarious effects on systems doing just that.

But even if you had a practical preimage attack on MD5—say, an array of quantum computers large enough to run Grover's algorithm—it would give you a preimage, but not necessarily (and probably not) the preimage I have in mind that you'll find with a web search.  Instead, it would probably give you 128 or 192 or so bits of gibberish.  And if you ran it for longer it would probably give you many such blocks of gibberish, long before finding anything that resembles English text.

If the messages you want to transmit cover only a small subset of all bit strings, and are structured with a distribution you know something about, you could invent a better compression scheme for it.  Maybe if there are few enough messages with convenient enough structure, you could compress them into 128 bits.  This depends on specific information about the distribution of messages, and it depends on having no more than $2^{128}$ pigeons to stuff into holes.

But a preimage-resistant cryptographic hash function like MD5 would be about the stupidest compression scheme imaginable, because not only was it designed without knowledge of your distribution of messages, but it goes to great lengths to obscure the original message and to make ‘decompression’ as hard as it can so that the best way is to guess a possible original message and check it.