  Associative memories are devices storing information that can be fully
retrieved given partial disclosure of it. We examine a toy model of associative
memory and the ultimate limitations it is subjected to within the framework of
general probabilistic theories (GPTs), which represent the most general class
of physical theories satisfying some basic operational axioms. We ask ourselves
how large the dimension of a GPT should be so that it can accommodate $2^m$
states with the property that any $N$ of them are perfectly distinguishable.
Call $d(N,m)$ the minimal such dimension. Invoking an old result by Danzer and
Gr\"unbaum, we prove that $d(2,m)=m+1$, to be compared with $O(2^m)$ when the
GPT is required to be either classical or quantum. This yields an example of a
task where GPTs outperform both classical and quantum theory exponentially.
More generally, we resolve the case of fixed $N$ and asymptotically large $m$,
proving that $d(N,m) \leq m^{1+o_N(1)}$ (as $m\to\infty$) for every $N\geq 2$,
which yields again an exponential improvement over classical and quantum
theories. Finally, we develop a numerical approach to the general problem of
finding the largest $N$-wise mutually distinguishable set for a given GPT,
which can be seen as an instance of the maximum clique problem on $N$-regular
hypergraphs.
