Difference between time-bounded and memory-bounded Kolmogorov complexity
Assume that there exists a sparse set $L \in \mathbf{NP} -\mathbf{P}$, this is equivalent to $\mathbf{EXP}\not= \mathbf{NEXP}$.

Then we can construct such a sequence. Indeed, consider $L_n = \{l_1,\ldots, l_k\} = L \cap \{0,1\}^n$.

Denote by $s_i$ the lexicographically first certificate for $l_i$. 

Define $x_n$ as the list of pairs: $ x_n:= \{ (l_1, s_1), \ldots, (l_k,s_k)\}$. 

Certainly, $C_{\mathsf{poly}(n)}(x_n) = O(\log n)$. Let us show that $C^{\mathsf{poly}(n)}(x_n) &gt; c\log n$ for every $c$.

Indeed, otherwise there exists a polynomial-time algorithm calculating   all strings of complexity  at most $c\log n$ (including $x_n$). Hence we can verify that a string belongs to $L$ in polynomial time.