Frankly all of them seems to be invented privately, and then gets backed by the state as a last resort when private parties gets into a dispute and risk taking a large part of the economy down with them.
And apparently the notion of debt exited long before the notion of money. And even afterwards money was something used between individuals that didn't know each other (different tribes effectively).
Some think that money in the form of shells existed back to at least 75,000 B.P.[1]. Language and controlled fire arose around the same time. Money (easily transferable anonymous debt) may have been used by (proto?)humans before the notion of words. Money may have been one of the few keys to the rise of human dominance.
And apparently the notion of debt exited long before the notion of money. And even afterwards money was something used between individuals that didn't know each other (different tribes effectively).