Money is chosen because money is convenient. Current prevailing opinions suggest a four major factors that affect the choice of a money:
- Durability: The currency should avoid devaluing over time
- Divisibility: It should be possible to use the currency for arbitrarily small transactions.
- Transportability: It should be easy to transport a great deal of wealth
- Noncounterfitability: It should be hard to create fake money.
In your scenario, the last hold-out of the banks, quantum cryptography, fell, leaving non-counterfitability no longer feasible. You want to know where it goes.
I'm assuming by your "apocalypse" tag that we should make this worse than the question actually asks for. Let's say QM crypto took a long time to break, and along the way we figured out how to make gold from carbon, etc. Lets drag the concept of currency all the way back to its basics.
Money is funny in that its primary purpose is trade. However, there is no reason its ONLY purpose is trade. Several cultures in the Americas used Cacao as currency. Yes, in that era, money actually did grow on trees. In some cultures, it was even so valuable that the peasants could not afford to drink hot chocolate -- it was a noble's drink simply because it cost too much to prepare!
This shows the simplest layer of money: an object that can be used by a sufficiently large portion of the population as to develop value. This is known as "commodity money." Cigarettes in POW camps is another common example of this sort of money. Even those who did not smoke accepted cigarettes as currency because they knew they could use them to trade with those who did smoke.
In this incredibly perverse apocalyptic world, there is one thing which should still hold value: energy. If we have enough energy capabilities to uproot Gold from its platform, we have certainly learned a thing or two about storing energy. Large powerful batteries would easily form the basis for a energy based currency. They pass all of the standards:
- Durability - Large batteries in this highly technological world would not deplete easily.
- Divisibility - Limited only by quantum particle limitations, energy is virtually perfectly dividable across any number of stores.
- Transportability - We're handwaving this until the next argument.
- Non-counterfitability - Nobody has figured out how to counterfit a Joule yet.
But what if energy is hard to transport? It would be a strange apocalypse where energy is hard to transport but gold is forgable: this might even qualify as a utopia to some people. Accordingly, I'm going to assume that either batteries or gold is a valid form of storage.
Once you have a physical good like a battery or gold, banks can come back into play. Right now we accept that we can put a dollar into bank A and withdraw it a thousand miles away in bank B. Crush all of our crypto algorithms, and this is no longer a safe assumption. Accordingly, each bank would have to be responsible for its own reserves, gold or energy alike. It would issue a certificate for each deposit. Bank deposits are tied to their physical location, and have to be moved by bank order if you wish to withdraw them elsewhere.
These certificates are safe, even in the presence of QM crypto breaking capabilities. They can function as a shared secret, with a random string of characters identifying each certificate. This shared secret is invulnerable to QM because owners of these certificates are advised to never place the strings in a hackable storage medium -- a physical safe would be best. Both the owner and the bank know the shared secret, but this secret is never transmitted until the withdraw is made. No cryptography can break such a system.
The resurgence of trust frees banks from this physical good trap. To make a withdaw from a far-off bank, you would have to give the local bank your certificate, and have a new one issued which says "in X days, when the banks re-distribute goods, this certificate is good for Y energy/gold." You then have to sit and trust the banks... but we rather trusted them already.
There is a funny aspect to these certificates, which is that they are easy to transfer with a little trust. If you trust that a certificate is not counterfitted, you can simply assume ownership of the certificate, and withdraw the goods at a future date. If you trust that a bank will actually transfer the resources, you can even accept one of those futures certificates used to tranfer resources.
This will rapidly lead to a trust-based economy, where it is possible to forge a certificate, but it is rare for this to occur because the entire culture values the simplicity of being able to trust certificates. With this step, you have boot-strapped the gold-standard, for you will have pieces of paper which have a promised value in gold or energy.
Think that utopian ideal is unlikely? Guess again. We humans are far more creative than most recognize. Let's shave a little off of the durability of the certificates every time we trade them: let's say 0.11% of the transaction cost. In exchange for that durability, we can fund a community whose sole job is managing the currency for us. When a transaction is fraudulent due to a poor certificate, they will investigate. The merchant will usually be held responsible, because they have the most ability to cope with such a burden, but sometimes the community absorbs the cost. Consumers can feel safe because they know that, even though the merchant may hold a set of numbers which can make any arbitrary charge they please, merchants don't. If they did, the community would find out, and revoke their membership in the community.
- Durability: The value of the transaction only depreciates 0.11% each transaction
- Divisibility: The value is represented as a number which is written down, so it is arbitrarily divisible.
- Transportability: Only a few numbers are needed on one's person to have access to their entire life worth
- Noncounterfitability: There is an entire community dedicated to ensuring this is not an issue. It happens from time to time, but the system absorbs such costs.
This system is called Visa, or MasterCard. It is in use today.
Welcome to the apocalypse.