The following is a pretty easy consequence of the Macaulay bound, in the commutative case. Could anything like this be true in the non-commutative case?
Let $h(n)$ be a numerical polynomial. Then there exists an integer $r>1$, such that the following holds:
Suppose $A$ is a graded $\mathbb{C}$-algebra, such that $A_0=\mathbb{C}$, and $\dim A_i=h(i)$, for all $1\leq i \leq q$, for some $q\gg 0$. Assume also that $A$ is generated in degree $1$ as $\mathbb{C}$-algebra, and it admits relations all of whose degrees are $\leq q$. Then, in fact, $A$ it admits relations which are in degrees $\leq r$.
Thanks for any hints.
