  This is the second part in a series of papers on counting surfaces on
Calabi-Yau 4-folds. In this paper, we introduce $K$-theoretic $\mathrm{DT},
\mathrm{PT}_0, \mathrm{PT}_1$ invariants and conjecture a
$\mathrm{DT}$-$\mathrm{PT}_0$ correspondence. For certain tautological
insertions, we derive Lefschetz principles in both the compact and toric case
allowing reductions to 3-dimensional $\mathrm{DT}, \mathrm{PT}$ invariants. We
also develop a topological vertex and conjecture a
$\mathrm{DT}$-$\mathrm{PT}_0$ vertex correspondence. These methods enable us to
verify our conjectures in several examples.
