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The equation of state (EoS) of QCD is a crucial input for the modeling ofheavy-ion-collision (HIC) and neutron-star-merger systems. Calculations of thefundamental theory of QCD, which could yield the true EoS, are hindered by theinfamous Fermi sign problem which only allows direct simulations at zero orimaginary baryonic chemical potential. As a direct consequence, the currentcoverage of the QCD phase diagram by lattice simulations is limited. In theseproceedings, two different equations of state based on first-principle latticeQCD (LQCD) calculations are discussed. The first is solely informed by thefundamental theory by utilizing all available diagonal and non-diagonalsusceptibilities up to $\mathcal{O}(\mu_B^4)$ in order to reconstruct a fullEoS at finite baryon number, electric charge and strangeness chemicalpotentials. For the second, we go beyond information from the lattice in orderto explore the conjectured phase structure, not yet determined by LQCD methods,to assist the experimental HIC community in their search for the criticalpoint. We incorporate critical behavior into this EoS by relying on theprinciple of universality classes, of which QCD belongs to the 3D Ising Model.This allows one to study the effects of a singularity on the thermodynamicalquantities that make up the equation of state used for hydrodynamicalsimulations of HICs. Additionally, we ensure that these EoSs are valid forapplications to HICs by enforcing conditions of strangeness neutrality andfixed charge-to-baryon-number ratio.

Lattice-QCD-based equations of state at finite temperature and density