A polynomial geometric progression with small-degree terms

Question

Suppose that $p$ is a large prime, and $Q_1,\dotsc,Q_p,P\in\mathbb F_p[x]$ are non-zero polynomials such that for every $i\in[1,p]$, the polynomial $Q_{i+1}$ is the remainder of division of $Q_iP$ by $x^p-x$ (here $Q_{p+1}=Q_1$). Given that the degrees of the polynomials satisfy $\deg Q_i\le p-1$ and $\deg P\le p$, and in addition $\deg Q_i\le 0.9p+i$ for $i\le 0.1p$, can one conclude that $\deg P\le 10$, or something else of this sort?

Answer 1

How about taking $Q_i = x$ for all $i$ and $P = x^{p-1}$?

This satisfies all your requirements and works for all primes (except maybe really small ones), but you can't bound the degree of $P$.