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The equation of state (EoS) of quantum chromodynamics (QCD) at zerotemperature can be calculated in two different perturbative regimes: for smallvalues of the baryon chemical potential $\mu$, one may use chiral perturbationtheory (ChEFT); and for large values of $\mu$, one may use perturbative QCD(pQCD). There is, however, a gap for $\mu \in (0.97\text{ GeV},\, 2.6\text{GeV})$, where these theories becomes non-perturbative, and where there iscurrently no known microscopic description of QCD matter. Unfortunately, thisinterval obscures the values of $\mu$ found within the cores of neutron stars(NSs).  In this thesis, we argue that thermodynamic matching of the ChEFT and pQCDEoSs is a legitimate way to obtain quantitative constraints on thenon-pertubative QCD EoS. Moreover, we argue that this method is effective,verifiable, and systematically improvable. First, we carry out a simplifiedmatching procedure in QCD-like theories that can be simulated on the latticewithout a sign problem. Our calculated pressure band serves as a prediction forlattice-QCD practitioners and will allow them to verify or refute thesimplified procedure. Second, we apply the state-of-the-art matched EoS ofKurkela et al. (2014) to rotating NSs. This allows us to obtain bounds onobservable NS properties, as well as point towards future observations thatwould more tightly constrain the current state-of-the-art EoS band. Finally, asevidence of the ability to improve the procedure, we carry out calculations inpQCD to improve the zero-temperature pressure. We calculate the full$\mathcal{O}(g^{6} \ln^{2} g)$ contribution to the pQCD pressure for $n_{f}$massless quarks, as well as a significant portion of the $\mathcal{O}(g^{6} \lng)$ piece and even some of the $\mathcal{O}(g^{6})$ piece.

From pQCD to neutron stars: matching equations of state to constrain global star properties