Can a state deny the right to vote or education to blacks?
Depends.
Are they liberals?
Constitutionally a state cannot have laws that segregate blacks. Other than that, state laws are up to the people of that state. States are considered "local governance" so your voice is much more effective at the state level than at the national level.
Nothing is perfect, but decentralized and local governance is the best way to ensure enduring freedom from government coercion.
If you and yours have tried and still have problems with your state, you can always move to another state that is more to your liking.