Quantum $j$-Invariant in Positive Characteristic II: Formulas and Values at the Quadratics

The multi-valued quantum $j$-invariant in positive characteristic is studied at quadratic elements. For every quadratic $f$, an explicit expression for each of the values of $j^{\rm qt}(f)$ is given as a limit of rational functions of $f$. It is proved that the number of values of $j^{\rm qt}(f)$ is finite and that none of these values is $\infty$.