Summary: We study two classes of quadratic algebras over a field \(k\): the class \(\mathfrak C_{n}\) of all \(n\)-generated PBW algebras with polynomial growth and finite global dimension, and the class of quantum binomial algebras. We show that a PBW algebra \(A\) is in \(\mathfrak C_{n}\, iff\) its Hilbert series is \(H_{A}(z)=1/(1 - z)^n\). Furthermore, the class \(\mathfrak C_{n}\) contains a unique (up to isomorphism) monomial algebra, \(A=k\langle x_1,\dots ,x_n\rangle /(x_j x_i| 1\leq i< j \leq n)\). A surprising amount can be said when \(A\) is a quantum binomial algebra, that is its defining relations are nondegenerate square-free binomials \(xy-c_{xy}zt,c_{xy}\in k^\times\). Our main result shows that for an \(n\)-generated quantum binomial algebra \(A\) the following conditions are equivalent: (i) \(A\) is an Artin-Schelter regular PBW algebra. (ii) \(A\) is a Yang-Baxter algebra, that is the set of relations \(\mathfrak R\) defines canonically a solution of the Yang-Baxter equation. (iii) \(A\) is a binomial skew polynomial ring, with respect to an enumeration of \(X\). (iv) The Koszul dual \(A^{!}\) is a quantum Grassmann algebra.