Name one that actually has worked in recorded history.
The problem with digital currency is that some central body needs to keep track of who has what. That's way too much centralized power.
As I said, Bitcoin and its derivatives are one such example. The scarcity of the new coins is a well known fact, if the mathematical proof does not suffice.
The problem with digital currency is that some central body needs to keep track of who has what.
There is no "central body" in the Bitcoin system. Some say it's also a drawback, for a number of reasons. There doesn't need to be a central body in other systems either. A true cash replacement can't even have a mandatory central body because that would imply 100% connection to some database. This isn't an option if you are buying peaches from a farmer on the side of a road in the middle of nowhere.
Here is an example that I spent not more than a minute inventing. It's far from perfect. But here it is, a "Hello, World!" digital currency.
Take a number, from zero to a trillion. Concatenate a random number with it. Digitally sign the result. What you have now is a newly minted digital coin. You can check the signature on it at any online location and verify that it is valid.
The bank will hold and manipulate this currency just as any other; so if you are using debit or credit cards you are not affected at all. The changes are at the cash transaction level, so let's have a look at how that works.
If you want to work with cash you need a digital wallet. It contains coins. The wallet is protected with any measures you want (PIN code, fingerprint, voice, whatever) so if stolen it will be useless to the thief. If found, you can claim ownership by successfully unlocking it.
Two wallets can securely and atomically transfer coins from one to another. This is done by a handshaking protocol, during which wallets exchange their identities (not identities of participants, though - you are still anonymous). If one coin leaves the wallet A then it arrives in wallet B, otherwise the transaction doesn't take place. The physical layer of communication can be anything you want - wired or wireless.
Since every coin is individually numbered, wallets can keep track of transfers, and they can be - if desired - periodically synchronized with a large distributed database. This allows you to keep track of your payments, and of payments made to you. You don't have to publish anything; but even if you do, only requests about *exact* coin numbers will be answered, and only you and the other guy know them. A third party can't guess those numbers because of the random salt in the coin number. The system will also restrict the tracking to a transaction done using your wallet.
I constructed this system specifically to address a serious problem of Bitcoin - the workload on the servers. Bitcoin doesn't scale, and at the same time it depends on everyone publishing their transactions (so that bitcoins can't be duplicated.) My scheme doesn't require publishing of anything; but instead it has another weakness - the wallet must be secure (tamper-proof) and it can't be backed up. That means that if you lose the wallet you lose all the cash that is inside.
As I said, this is a quick and dirty example. But even that scenario was used in several SciFi novels; people would wear tamper-proof bracelets, and they would contain credits, and credits would be transferrable to other bracelets (or terminals) as instructed by the owner.
Large purchases still could be handled by banks through transfers. In today's society cash is often an inconvenience (it generates all those coins...) But if in the future banks will not be present then those digital wallets can be issued as your sole storage of money. Of course you can have more than one wallet, and you can transfer money between them just as easily as you do it with cash.
Wallets can be made to be hidden in houses, so that they are inaccessible to thieves (especially if they are wireless.) You don't need an accessible safe anymore; all you need is a brick in a wall that is drilled out during the construction of the house and a blank digital wallet is inserted. You can then paint the wall. Whenever you need money you come close to that brick, enter codes, and the money goes in or out. If the in-wall wallet uses RFID it doesn't need its own power and can exist forever; but all wallets will have NVRAM for storage, and lots of forward error correction, so they are nearly bulletproof.
Counterfeiting will be prevented primarily by the secure wallets. They can be manufactured to be physically impossible (or very difficult) to duplicate or to open up and retrieve the keys. But if someone manages to do that and copy coins, this will result in existence of the same coin in several wallets. You may not want to publish your transactions, but eventually the coin makes its way to a place that publishes them, and then the problem coin will be found. If you are anonymous nobody can get to you this way; however if you pay at commercial places they may do online checks on your coins, and if that coin is known to exist somewhere else in the system (the central database that you are talking about!) then your transaction will be flagged and you will have some explaining to do. Note that the central database here is an optional feature, and nobody is required to use it. If you are concerned about privacy, paper bills are also numbered, and in theory they can be traced from the moment the bank issued them to you to the moment you spent them.
I think digital currency is unavoidable. We can't depend on physical materials (like gold) because of a million reasons. A good digital currency is more resistant to forgery and can't be issued excessively. Anyone who has a coin can check it against the issuer's numbering scheme and if the issuer minted one too many it will be a red flag right there.