Protesters in the towns of Rohtak and Jind set fire to police vehicles, public buildings and buses.
The demonstrators are mostly from the Jat community who are unhappy about India's caste quota system.
They say it puts them at a disadvantage in government jobs and at state-run educational institutes.
The Jats are currently listed as upper caste but the demonstrators want job quotas similar to those granted to lower castes.
Protesters again went on the rampage on Saturday despite a curfew and the deployment of the army which is reported to have opened fire on them in the Rohtak and Jhajjar districts.
Demonstrations by Jats were also reported to have taken place in Delhi, Punjab, Himachal Pradesh and Jammu and Kashmir. Overland transport links to Haryana have been brought to a halt by the protests.
The government of Haryana has meanwhile been urged by the central government to initiate talks with Jat leaders, NDTV reported.
Haryana Chief Minister Manohar Lal Khattar has already said that the government is ready to agree to the demands of the protesters "within the ambit of the constitution".
Mr Khattar has urged the protesters to desist and appealed to the government to provide more troops in order to restore order.
Opposition parties have called for Mr Khattar's government to be sacked by the central government and for presidential rule to be imposed on the state.
On Friday protesters in Rohtak hurled rocks at security forces, while blocking traffic, attacking vehicles and attempting to set the finance minister's home on fire.
At least five people have been killed in a second day of violence related to caste rights in the northern Indian state of Haryana, police say.