Quantum Money where not even the Bank can counterfeit
There are proposals for quantum money where it appears that not even the bank can produce two copies of a quantum money state with the same serial number. See 


Farhi et al's paper Quantum Money from Knots, 
Mark Zhandry's paper Quantum Lightning Never Strikes the Same State Twice,  
Daniel Kane's paper Quantum Money from Modular Forms.


In order to make sure that the bank can only produce $k$ banknotes, one would need to have the bank publish the serial numbers of exactly $k$ banknotes in some secure location. Otherwise it could make as many bank notes as it wants (although they would all have different serial numbers).