How is heat represented on a quantum level?

Heat energy, at a microscopic level, is stored in the degrees of freedom of atoms and molecules. These degrees of freedom are translational, rotational and vibrational. They all store different amounts of energy, depending on the geometry of the atom. Translational degrees of freedom are the atom or molecule moving around in space, and there are always 3 for the 3 dimensions of space. The rotational and vibrational modes come from the geometry of the atom/molecule.

From quantum mechanics, we get the idea that energy stored in rotational and vibrational (and translational, if confined) modes must come in quantised packets, with a minimum size. This size depends on the form of a certain mode. For single atoms, the moment of inertia and the energy of rotation is very small. The quantum of energy that must be added to excite the rotational modes is large, and so these do not contribute to heat storage until very high temperatures.

Molecules have much higher moments of inertia around certain axes. For example, O2 has high moment of inertia around the two axes perpendicular to its bond axis and a low moment of inertia around its bond axis. It therefore stores heat energy in those two and they contribute to the heat capacity of O2.

Vibrational modes store much more energy than translational or rotational modes, and are active only at higher temperatures.

This is basically what heat is at a microscopic level. Quantum mechanics gives us that the energy stored in the modes must be quantised.

The equipartition theorem says that all modes of excitation carry heat. There may be some modes which are too energetic to be excited at a given temperature, but the remaining modes are all excited. In overly simple terms, everything that can shake will shake.