QCD is well understood at short distances where perturbative calculations are feasible. Establishing an explicit analytic connection between the short-distance regime and the large-distance physics of quark confinement has been a long-sought goal. A major challenge is to relate the scale \Lambda_{s} underlying the evolution of the QCD coupling in the perturbative regime to the masses of hadrons. We show here how new theoretical insights into the behavior of QCD at large distances leads to such a relation. The resulting prediction for \Lambda_{s} in the \overline{MS} scheme agrees well with experimental measurements. Conversely, the relation can be used to predict the masses of hadrons composed of light quarks with the measured value of \Lambda_{s} as the sole parameter. We also use "light-front holography" to determine the analytic form of \alpha_s(Q^2) at small Q^2. Show Partial Abstract