Tasse,
UI is provided by the state (though paid thru the employer's insurance contributions).
Q6. (3) State and local cash assistance programs for income maintenance (often called state "General Assistance," but which may exist under other names).
General assistance and UI are 2 different things but if someone is ineligible for UI they could be for TANF/GA. Also, be aware that if UI is delayed, one can be provided GA/TANF.
In mpltpl's case, if the project ends (could be the end of a mutual contractual agreement), may not be eligible for UI. But, if it is layoff or 'forced' to quit, then eligible for UI. But, for the primary applicant the basis of GC (for emploment based) is on the job and hence claiming UI will be extremely risky.
I also read (I will try to get the link) that someone claiming more than 12 months of UI (not sure of the overall period, say within 5 years, 4 years, etc.) is 'deemed' a public charge. So, if someone repeatedly claims UI to hit 12 months, then there is a risk of INS getting the notification of the public charge.
As always checking with the state and/or a lawyer makes better sense.