Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both an algebra and coalgebra grading, with $K(0)=k$). Call such a Hopf algebra a connected Hopf-graded Hopf algebra.

Connected Hopf-graded Hopf $k$-algebras arise naturally in algebraic toplogy when studying the cohomology rings (with coeffecients in $k$) of $H$-spaces. I assume (although I'm not 100% sure), that not all such Hopf algebras arise as cohomology algebras in this way. My question is therefore the following:

Let $K$ be a connected Hopf-graded Hopf algebra. Which additional properties on $K$ guarantee that it can be viewed as the cohomology ring $K^{*}(X;k)$ of some $H$-space $X$?