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Given a prime 
$p\geqslant 5$ and an integer 
$s\geqslant 1$, we show that there exists an integer 
$M$ such that for any quadratic polynomial 
$f$ with coefficients in the ring of integers modulo 
$p^{s}$, such that 
$f$ is not a square, if a sequence 
$(f(1),\ldots ,f(N))$ is a sequence of squares, then 
$N$ is at most 
$M$. We also provide some explicit formulas for the optimal 
$M$.
