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The equation of state (EoS) of QCD is a crucial input for the modeling of heavy-ion-collision (HIC) and neutron-star-merger systems. Calculations of the fundamental theory of QCD, which could yield the true EoS, are hindered by the infamous Fermi sign problem which only allows direct simulations at zero or imaginary baryonic chemical potential. As a direct consequence, the current coverage of the QCD phase diagram by lattice simulations is limited. In these proceedings, two different equations of state based on first-principle lattice QCD (LQCD) calculations are discussed. The first is solely informed by the fundamental theory by utilizing all available diagonal and non-diagonal susceptibilities up to O(µ 4 B) in order to reconstruct a full EoS at finite baryon number, electric charge and strangeness chemical potentials. For the second, we go beyond information from the lattice in order to explore the conjectured phase structure, not yet determined by LQCD methods, to assist the experimental HIC community in their search for the critical point. We incorporate critical behavior into this EoS by relying on the principle of universality classes, of which QCD belongs to the 3D Ising Model. This allows one to study the effects of a singularity on the thermodynamical quantities that make up the equation of state used for hydrodynamical simulations of HICs. Additionally, we ensure that these EoSs are valid for applications to HICs by enforcing conditions of strangeness neutrality and fixed charge-to-baryonnumber ratio. 

The Beam Energy Scan program at the Relativistic Heavy Ion Collider (RHIC) is searching for the QCD critical point. The main signal for the critical point is the kurtosis of the distribution of proton yields obtained on an event by event basis where one expects a peak at the critical point. However, its exact behavior is still an open question due to out-of-equilibrium effects and uncertainty in the equation of state. Here we use a simplistic hydrodynamic model that enforces strangeness-neutrality, selecting trajectories that pass close to the critical point. We vary the initial conditions to estimate the effect of out-of-equilibrium hydrodynamics on the kurtosis signal.