Dependence of entropy and temperature on Planck's constant

[spaghetti3451]

Consider the dependence of entropy and of temperature on the reduced Planck's constant (taken from page 23 of Thomas Hartman's lecture notes(http://www.hartmanhep.net/topics2015/) on Quantum Gravity):

$$S \propto \hbar, \qquad \qquad T \propto \hbar.$$

I do not quite see how entropy can depend on the reduced Planck's constant. To give credence to my claim, consider the definition of (statistical) entropy in classical and in quantum systems.

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In a classical system, the (statistical Gibbs) entropy for a macroscopic system (with a discrete set of microstates) is defined as

$$S = -k_\text{B}\,\sum_i p_i \ln \,p_i,$$

where $k_{\text{B}}$ is Boltzmann's constant and $p_i$ is the probability that the system is in microstate $i$ during the system's fluctuations.

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In a quantum system, the (statistical von Neumann) entropy for a macroscopic system (with a discrete set of microstates) is defined as

$$S = -k_\text{B}\,\text{Tr}\ (\rho \ln \rho),$$

where $k_{\text{B}}$ is Boltzmann's constant and $\rho$ is the density matrix of the system.

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How then does entropy depend on the reduced Planck's constant in quantum systems?

Why should temperature also depend on the reduced Planck's constant in quantum systems?

 

[Jilang]

I can see how the number of distinguishable miscrstates might be related to the inverse of h, but not directly proportional to it. (As h tends to zero the number of micro states would to infinity.)