The statement is false. For example, for the case of $d=1$, it is known that the largest number of consecutive quadratic residues modulo $p$ tends to infinity as $p\to\infty$. See here and, for more general patterns, see Seva's answer to this MO question. Further related MO entries are here and here.
Added. Seva's answer (linked above) shows that the statement is false for every $d$.
