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While the main body of the constitution does not lay out any requirements for the vice-presidency, the amendments most certainly do.
The final paragraph of the twelfth amendment reads:
But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
Therefore, the Vice-President must be constitutionally eligible for the Presidency. These requirements are that they must be a natural born citizen of the United States, that they be no less than thirty five years old, and that they have been a resident of the United States for at least fourteen years.
For this reason, somebody such as Arnold Schwarzenegger would not be eligible to run as Vice President, because while he is over thirty five and has lived in the USA for longer than fourteen years, he was not born a US citizen