In a joint work with R. Benedetti, we recently defined a new
family of quantum invariants, the "Quantum Hyperbolic Invariants" (QHI),
for closed $3$-manifolds endowed with flat principal bundles, which
generalize the colored Jones polynomials of links in $S^3$. In this
talk, we present  the fundamental geometric concepts in the construction
of QHI, and then discuss their interpretation in terms of scissors
congruence classes of $3$-manifolds. For hyperbolic $3$-manifolds, this
interpretation leads to a natural motivationfor the Volume Conjecture,
which predicts that the asymptotic behaviour of QHI would eventually
recover the volume and the Chern-Simons invariant.