  We provide a homological model for a family of quantum representations of
mapping class groups arising from non-semisimple TQFTs (Topological Quantum
Field Theories). Our approach gives a new geometric point of view on these
representations, and it gathers into one theory two of the most promising
constructions for investigating linearity of mapping class groups. More
precisely, if $\varSigma_{g,1}$ is a surface of genus $g$ with $1$ boundary
component, we consider a (crossed) action of its mapping class group
$\mathrm{Mod}(\varSigma_{g,1})$ on the homology of its configuration space
$\mathrm{Conf}_n(\varSigma_{g,1})$ with twisted coefficients in the Heisenberg
quotient $\mathbb{H}_g$ of its surface braid group
$\pi_1(\mathrm{Conf}_n(\varSigma_{g,1}))$. We show that this action intertwines
an action of the quantum group of $\mathfrak{sl}_2$, that we define by purely
homological means. For a finite-dimensional linear representation of
$\mathbb{H}_g$ (depending on a root of unity $\zeta$), we tweak the
construction to obtain a projective representation of
$\mathrm{Mod}(\varSigma_{g,1})$. Finally, we identify, by an explicit
isomorphism, a subrepresentation of $\mathrm{Mod}(\varSigma_{g,1})$ that is
equivalent to the quantum representation arising from the non-semisimple TQFT
associated with quantum $\mathfrak{sl}_2$ at $\zeta$. In the process, we
provide concrete bases and explicit formulas for the actions of all the
standard generators of $\mathrm{Mod}(\varSigma_{g,1})$ and of quantum
$\mathfrak{sl}_2$ on both sides of the equivalence, and answer a question by
Crivelli, Felder, and Wieczerkowski. We also make sure that the restriction of
these representations to the Torelli group $\mathcal{I}(\varSigma_{g,1})$ are
integral, in the sense that the actions have coefficients in the ring of
cyclotomic integers $\mathbb{Z}[\zeta]$, when expressed in these bases.
