  Nonperturbative methods play an important role in quantum many-body systems,
especially in situations with an interplay of continuum and bound states and/or
large coupling strengths between the constituents. Employing the Luttinger-Ward
functional (LWF) we have computed the equation of state (EoS) of the
quark-gluon plasma (QGP) using fully dressed selfconsistent 1- and 2-body
propagators. We first give an alternative derivation of our previously reported
results for resumming the ladder diagram series of the LWF using a "matrix log"
technique which accounts for dynamically formed bound and resonant states. Two
types of solutions were found in selfconsistent fits to lattice-QCD data for
the EoS, heavy-quark free energy and quarkonium correlators: a strongly coupled
scenario (SCS) with broad parton spectral functions and strong meson resonances
near the transition temperature vs. a weakly coupled scenario (WCS) with
well-defined parton quasiparticles and weak meson resonances. Here, we discuss
how these solutions can be distinguished by analyzing the pertinent transport
properties. We focus on the specific shear viscosity, $(4\pi )\eta/s$, and the
heavy-quark diffusion coefficient, $(2\pi T) {\cal D}_s$, including its mass
dependence. At low temperatures, in the SCS, they turn out to be a factor of 2
within their conjectured quantum lower bound, while they are a factor of 2-5
larger in the WCS. At higher temperatures, the transport parameters of the two
scenarios approach each other. We propose the ratio $ (4\pi\eta/s)/(2\pi T
{\cal D}_s )$ as a measure to distinguish the perturbative and strong-coupling
limits of 5/2 and 1, respectively.
