  In this work we study the quantum periods together with their Picard-Fuchs
differential equations of Calabi-Yau fourfolds. In contrast to Calabi-Yau
threefolds, we argue that the large volume points of Calabi-Yau fourfolds
generically are regular singular points of the Picard-Fuchs operators of
non-maximally unipotent monodromy. We demonstrate this property in explicit
examples of Calabi-Yau fourfolds with a single Kahler modulus. For these
examples we construct integral quantum periods and study their global
properties in the quantum Kahler moduli space with the help of numerical
analytic continuation techniques. Furthermore, we determine their genus zero
Gromov-Witten invariants, their Klemm-Pandharipande meeting invariants, and
their genus one BPS invariants. In our computations we emphasize the features
attributed to the non-maximally unipotent monodromy property. For instance, it
implies the existence of integral quantum periods that at large volume are
purely worldsheet instanton generated. To verify our results, we also present
intersection theory techniques to enumerate lines with a marked point on
complete intersection Calabi-Yau fourfolds in Grassmannian varieties.
