There is no text in the Quran which says that women can't become head of state. But there are ahadith which say that she can't become head of state.
Lets analyze whether it is advisable for a woman to be head of state or not.
If a woman is a head of a State in an Islamic country, she may have to lead the congregational prayer. And if a woman leads the congregational prayer; in Islam we have adoptable postures like Qiyam, Rukoo, Sujud i.e standing, bowing and prostration. And if a lady is doing that in front of a gents’ congregation, this would surely cause disturbances in the prayer.
If she happens to be the head of a modern society like the one we have today, many a times the head of the state has to hold meetings with other head of states which are mainly gents, and many a times they have close door meetings in which no one is allowed. And this Islam does not permit because we know that if a man and a woman are alone together, the third one between them is Satan. Islam does not permit intermingling of sexes.
For a head of state, it is necessary to meet the common man, and if the head of state is woman, she might not be able to meet common man and try to solve his problem.
And science tells us that a woman, during her menstrual period, undergoes certain behaviors, mental and psychological changes due to the release of sex hormones. And these changes surely disturb her in making the right decisions if she is the head of state.
Science also tells us that a woman has more verbal and vocal skills as compared to a man. And man has more ability to imagine things, to imagine future, to imagine a future project. And this ability is very important for a head of a state. A woman has been given an edge over the verbal and vocal skills which are required for motherhood.
A woman may get pregnant and she surely may require few months rest, so who will look after the state at this time?
She may have children. Her duty as a mother is far more important! And it is more practicable for a man who can do both duties; of a father as well as a head of state as compared to a woman if she has children.
So for these reasons, a woman should not be a head of state. But this does not mean that she can’t take part in making decisions! Woman has a right to vote; she has a right in law making.
At the time of treaty of Hudaybiya, Umme Salam (r.a) supported and guided the Prophet at a time when the whole Muslim community was disturbed. So surely a woman can help the man in making decisions about the State!