This application of quantum mechanics, to produce harder to counterfeit cash, never occurred to me. I guess it should have as digital cash based on classical computers has been bounded up in the same areas of interest as classical cryptography. Cryptography and cryptanalysis have been advanced as likely areas where quantum computers will exceed classical ones. While still unproven, researchers have a strong suspicion in particular that the hard to reverse classical operations of public key cryptography will be feasible, maybe even trivial, for quantum computers to undo.
As the MIT Technology Review article explains, a team at MIT has been building and breaking quantum cash systems for at least a little while now. The latest development is that they think they have a class of asymmetric calculations that might prove resistant to cracking by quantum computers. Roughly, the computation involves what reads like trying to determine if there is a continuous topological transform between two knots.
Their quantum cash is based on a new kind of asymmetry: that two identical knots can look entirely different. So while it may be easy to make either knot, it is hard to find a way to transform one into the other.
The article goes on to pinpoint the problem with devising such systems. It is easy to build a cipher or calculation that the creator cannot easily reverse but that doesn’t guarantee someone else won’t find a trivial way of doing so. What has helped that more than anything in classical crypto is the availability of cheap and powerful computing, something that is still quite a ways off for quantum computers.