  We present a unified approach to the thermodynamics of hadron-quark-gluon
matter at finite temperatures on the basis of a quark cluster expansion in the
form of a generalized Beth-Uhlenbeck approach with a generic ansatz for the
hadronic phase shifts that fulfills the Levinson theorem. The change in the
composition of the system from a hadron resonance gas to a quark-gluon plasma
takes place in the narrow temperature interval of $150 - 185$ MeV where the
Mott dissociation of hadrons is triggered by the dropping quark mass as a
result of the restoration of chiral symmetry. The deconfinement of quark and
gluon degrees of freedom is regulated by the Polyakov loop variable that
signals the breaking of the $Z(3)$ center symmetry of the color $SU(3)$ group
of QCD. We suggest a Polyakov-loop quark-gluon plasma model with
$\mathcal{O}(\alpha_s)$ virial correction and solve the stationarity condition
of the thermodynamic potential (gap equation) for the Polyakov loop. The
resulting pressure is in excellent agreement with lattice QCD simulations up to
high temperatures.
