This is an open research question and known as the ``Online Boolean Matrix-Vector Multiplication problem''. This problem reads as follows (see [1]): Given a binary $n \times n$ matrix $M$ and $n$ binary column vectors $v_1, \dots, v_n$, we need to compute $M v_i$ before $v_{i+1}$ arrives. (Notice that allowing the vectors to have real-values would just make the problem harder.)

Cleary, the naive algorithm (which just uses standard matrix-vector-multipliction) takes time $O(n^3)$ to solve this problem. There is a conjecture (see e.g. [1]) that this cannot be done truly faster than $O(n^3)$. (In more detail, this conjecture goes as follows:  There exists no truly subcubic algorithm, which solves the Online Boolean Matrix-Vector Multiplication Problem, i.e. there is no algorithm with running time $O(n^{3 - \varepsilon})$ for $\varepsilon > 0$).

It is known that Williams' algorithm solves this problem in time $O(n^3 / \log^2 n)$. See [2] for more details.

It would be a breakthrough in the area of conditional lower bounds, if one could prove or disprove (e.g. by improving Williams' algorithm) the above conjecture.

[1] Unifying and Strengthening Hardness for Dynamic Problems via an Online Matrix-Vector Multiplication Conjecture. by Henzinger, Krinninger, Nanongkai and Saranurak <br>
[ http://eprints.cs.univie.ac.at/4351/1/OMv_conjecture.pdf ]

[2] Matrix-vector multiplication in sub-quadratic time: (some preprocessing required). by Williams <br>
[ http://dl.acm.org/citation.cfm?id=1283383.1283490 ]