What is the relationship between natural rights and political legitimacy in Hobbes and Locke? What are our natural rights, and how are they related to assessing whether a state is legitimate or not?

John Locke and Thomas Hobbes had different views on the source of a state's power and its legitimacy to rule. However, both philosophers agreed that power rested in a government as the result of a social contract in which subjects cede certain liberties to the government in order to live in an orderly society.
Hobbes looked at civic power in a cold, almost scientific way. As he argued in The Leviathan, strong leaders have the natural right to keep their subjects in check. His view of natural rights was an outgrowth of his view of man in his "state of nature." In his natural state, Hobbes argued, a person is free to do anything that will grant him more power and resources. There are no natural laws governing a person's behaviors and actions. Only liberty exists, and a state of "war against all" applies to everyone (Leviathan XIII). Hobbes considered natural rights to be in conflict with the idea of natural law. There are no laws in a state of nature, he argued. To prevent the cruelty that exists in a state of nature "there must be some coercive Power to compel men equally to the performance of their Covenants" (Leviathan XV). In short, political power derives its legitimacy from instances when people agree to give it or, more commonly, are forced by a ruler to surrender their natural rights for laws. According to Hobbes, a state is legitimate if it can keep people from devolving into the chaos of their natural state.
Locke took a different view of people in their natural state. He saw the state of nature as a condition in which people made a social contract to live together within a system of shared values and laws. Everyone, he argued, was naturally entitled to the rights of life, liberty, and property. A government's foremost duty is to protect these natural rights. In order to do so, it must exist according to the "consent of the governed" (Two Treatises of Government XVIII). According to Locke, a government that does not ensure the natural rights of its people has no legitimacy to rule.
In a sense, Hobbes and Locke agree on the function of government. They both argue that a government's duty is to protect the well-being of its citizens. Where they differ is in its source of legitimacy. Hobbes contends that a government's most crucial function is to protect people from their own greed as it exists in a state of nature. Any absolute monarch can do this. Locke, on the other hand, argues that people should voluntarily give certain powers to their rulers in order to protect their natural rights.
http://www.nlnrac.org/earlymodern/hobbes

http://www.nlnrac.org/earlymodern/locke