  Understanding the relationship between mock modular forms and quantum modular
forms is a problem of current interest. Both mock and quantum modular forms
exhibit modular-like transformation properties under suitable subgroups of
$\rm{SL}_2(\mathbb Z)$, up to nontrivial error terms; however, their domains
(the upper half-plane $\mathbb H$, and the rationals $\mathbb Q$, respectively)
are notably different. Quantum modular forms, originally defined by Zagier in
2010, have also been shown to be related to the diverse areas of colored Jones
polynomials, meromorphic Jacobi forms, partial theta functions, vertex
algebras, and more.
  In this paper we study the $(n+1)$-variable combinatorial rank generating
function $R_n(x_1,x_2,\dots,x_n;q)$ for $n$-marked Durfee symbols. These are
$n+1$ dimensional multisums for $n>1$, and specialize to the ordinary
two-variable partition rank generating function when $n=1$. The mock modular
properties of $R_n$ when viewed as a function of $\tau\in\mathbb H$, with
$q=e^{2\pi i \tau}$, for various $n$ and fixed parameters $x_1, x_2, \cdots,
x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono
when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$;
by Bringmann, Garvan, and Mahlburg for $n\geq 2$ and $x_1=x_2=\dots=x_n=1$; and
by the first and third authors for $n\geq 2$ and the $x_j$ suitable roots of
unity ($1\leq j \leq n$).
  The quantum modular properties of $R_1$ readily follow from existing results.
Here, we focus our attention on the case $n\geq 2$, and prove for any $n\geq 2$
that the combinatorial generating function $R_n$ is a quantum modular form when
viewed as a function of $x \in \mathbb Q$, where $q=e^{2\pi i x}$, and the
$x_j$ are suitable distinct roots of unity.
