  It is well-known that the cohomology ring has a richer structure than
homology groups. However, until recently, the use of cohomology in persistence
setting has been limited to speeding up of barcode computations. Some of the
recently introduced invariants, namely, persistent cup-length, persistent cup
modules and persistent Steenrod modules, to some extent, fill this gap. When
added to the standard persistence barcode, they lead to invariants that are
more discriminative than the standard persistence barcode. In this work, we
devise an $O(d n^4)$ algorithm for computing the persistent $k$-cup modules for
all $k \in \{2, \dots, d\}$, where $d$ denotes the dimension of the filtered
complex, and $n$ denotes its size. Moreover, we note that since the persistent
cup length can be obtained as a byproduct of our computations, this leads to a
faster algorithm for computing it for $d>3$. Finally, we introduce a new stable
invariant called partition modules of cup product that is more discriminative
than persistent $k$-cup modules and devise an $O(c(d)n^4)$ algorithm for
computing it, where $c(d)$ is subexponential in $d$.
