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.

The motion signed by 24 MPs said fares in London are "the most expensive in the world" and since 2008 the cost of a single bus journey has jumped by 50%.
It comes as passengers saw on average a 4.2% increase in fares this January, across the Tube, buses and trams.
The mayor's office says "every penny" from fares is invested in improvements.
The motion, which is led by Eltham MP Clive Efford, said: "London commuter fares have become the most expensive in the world and are an ever increasing pressure on Londoners' household budgets while real wages are falling and bills are rising."
A zone one to six travel card is now Â£400 more expensive, the MPs note, in addition to a 56% rise in bus fares over the last five years.
The MPs also urged "the Mayor of London to use his discretion to freeze fares at RPI for 2014, easing the pressure on ordinary Londoners during the current cost of living crisis".
A spokesperson for mayor Boris Johnson, said: "Every penny from fares is invested in improving transport for Londoners, resulting in one of the biggest and most reliable transport systems in the world.
"He (the mayor) will make a decision on fares for 2014 later this year but has publicly stated that he is determined to bear down on them as much as possible."
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.
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Labour MPs in London have signed a Commons motion calling for transport fares in the capital to be frozen at the rate of inflation for 2014.