The shrinking of technological devices leads to the emergence of exotic heat conduction behaviors such as quantization of thermal conductivity, phonon tunneling, and ballistic conduction. Understanding and exploiting these quantum effects is crucial for advancing technologies such as thermal management and designing advanced materials in nanoscale systems. This research has focused on two topics: the possibility of constructing a device based on phonon tunneling and the quantum thermal conductivity of amorphous graphene, which shows quantum effects on room temperature due to strong carbon-carbon bonds. In doing so, we calculated the transmission coefficients using Green’s functions for both systems, and the Kubo-Greenwood method was used additionally for amorphous graphene. We worked in the harmonic limit since the scattering due to the material’s internal structure is the dominant scattering mechanism in disordered materials. Thermal conductivities were calculated using the Landauer formulation. For the distribution function in the Landauer formula, two different distribution functions, Bose-Einstein and Maxwell-Boltzman, were used to determine the quantum and classical thermal conductivities. A thermal chromator and a medium with a phononic gap were adjoined and placed between two thermal reservoirs to construct the phonon tunneling device. The dependency of transport properties on the gap system’s length is investigated. Results reveal the possibility of building such a device. Besides, the classical thermal conductivities of amorphous graphene are almost twice as high as the quantum thermal conductivity, which shows that quantum thermal conductivity determines the thermal properties in high Debye materials where phonon-phonon interactions are suppressed.