AFAIK a muslim (man and woman) is allowed to recite Quran from memory - independent of the state he is in... although there are some opinions stating that if men are in the state of janaabah they need to first make ghusl.
Regarding "women during period" reading it from a screen/mobile etc. there seems to be no "100% consensus"... some say you are allowed, others say you are not... for a detailed analysis see here and here.
Allah knows best.