Can quran be recited in the state of periods?
Where we are not touching Quran and only reading it?

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Yes, according to Ayatullah Seestani, you can read the Quran except for the those verses which include obligatory Sajdas, although it is not allowed to touch the verses in such a state.
According to some jurists, when one cleans themselves during the prayer time and recites the Holy Quran will earn the same reward as if one is praying.