  When defining the amount of additive structure on a set it is often
convenient to consider certain sumsets; Calculating the cardinality of these
sumsets can elucidate the set's underlying structure. We begin by investigating
finite sets of perfect squares and associated sumsets. We reveal how arithmetic
progressions efficiently reduce the cardinality of sumsets and provide
estimates for the minimum size, taking advantage of the additive structure that
arithmetic progressions provide. We then generalise the problem to arbitrary
rings and achieve satisfactory estimates for the case of squares in finite
fields of prime order. Finally, for sufficiently small finite fields we
computationally calculate the minimum for all prime orders.
