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Is it possible to create a messaging application and apply a quantum cryptography technique to encrypt the messages between two parties within the application?

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    $\begingroup$ _Post-_Quantum crypto? yes. Actual quantum key exchanges? no, they require specialized hardware and specialized connections betweent he endpoints. $\endgroup$ – SEJPM Jul 4 '18 at 17:17
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NO, if "messaging application" is software running on an stock consumer-grade computer or variant (including mobile phone, tablet): in any of its standard meanings, Quantum Cryptography requires specific hardware.

For Quantum Key Distribution (actually quantum privacy amplification), which is how quantum cryptography does encryption, you'd need a specific interface; an optical fiber running to the recipient or ultra-precise alignment towards the receiving device in line-of-sight (with no fog/smog); AND the inconvenience of exchanging initial secret keys in sealed envelopes or equivalents (see this). QKD is a working solution looking for problems, and failing at that.

Quantum Computers are currently about as common as manmade objects on Mars, and as useful to make or break practical cryptography. Quantum Computing is an immature technology, and it is uncertain when (and even if) it will metamorphose into something practical, in particular for cryptography. At even odds, I would bet that controlled fusion will be first (that would make the energy wasted on cryptocurrency mining a non-issue).

Among doable things with a sensible use of the word "Quantum", I only see "Quantum-Safe Encryption", or otherwise said "Encryption by Post-Quantum Crytography". That is encryption believed safe even from hypothetical quantum computers usable for cryptanalysis. Using the common AES-256-GCM qualifies, and is easy, but that's symmetric encryption. For the convenience of public-key cryptography, we'd have to pick something in the realm of Post-Quantum cryptography; the state of that nascent art is presently competing there.

Another semantically justifiable thing is bragging for a "Quantum Random Number Generator". The OS's cryptographically secure RNG likely qualifies (any TRNG or matter is ultimately quantum-based anyway), and if it was not it would be hard to disprove the claim that it is.

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