Different traces of quantum systems having the same classical limit

Many quantum systems may have the same classical limit. We argue that in the classical limit their traces do not necessarily converge with each other. The trace formula allows to express quantum traces by means of classical quantities as sums over periodic orbits of the classical system. To explain the lack of convergence of the traces we need quantum corrections to the classical actions of the periodic orbits. Four versions of the quantum baker map on the sphere serve as an illustration of this problem.