Can we recognize the difference of two strings of $2^n$ length using polynomial size strings?
There is no sequence of functions $f_1,...,f_{n^c}:\{0,1\}^{2^n}\rightarrow\{0,1\}^{n^d}$, such that for all $s_1\neq s_2\in\{0,1\}^{2^n}$ $\exists i\in\{1,...,n^c\} : f_1(s_1)\neq f_i(s_2)$.

Examine the function $F:\{0,1\}^{2^n}\rightarrow \prod\limits_{i=1}^{n^c}\{0,1\}^{n^d}$ defined by $F(s)=\left(f_1(s),f_2(s),...,f_{n^c}(s)\right)$.

The domain of $F$ is of cardinality $2^{2^n}$, while its range is of cardinality $\left(2^{n^d}\right)^{n^c}=2^{n^{c+d}}$. Since for large enough $n$ it holds that $2^{2^n}&gt;2^{n^{c+d}}$, $F$ is not one to one, and thus the condition is not satisfied.