  In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial
generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee
symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the
ordinary two-variable partition rank generating function. Since then, it has
been a problem of interest to understand the automorphic properties of this
function; in special cases and under suitable specializations of parameters,
$R_n$ has been shown to possess modular, quasimodular, and mock modular
properties when viewed as a function on the upper half complex plane $\mathbb
H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum
modular forms, defined by Zagier in 2010, are similar to modular or mock
modular forms but are defined on the rationals $\mathbb Q$ as opposed to
$\mathbb H$, and exhibit modular transformations there up to suitably analytic
error functions in $\mathbb R$; in general, they have been related to diverse
areas including number theory, topology, and representation theory. Here, we
establish quantum modular properties of $R_n$.
