  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 $\mathcal{O}(\mu_B^4)$ 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-baryon-number ratio.
