Knowledge of Congress is low: Only 36 percent know who controls each chamber.
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The secretary of State faced questions on Turkey and Qatar.
Gillibrand's office declined to confirm or deny the report.
The potential changes are in response to a fence jumper reaching the building on Friday.
Omar J. Gonzalez, 42, reportedly had to retire because of a disability.
“I know there are contracts but this has gone on too long," she said.
"He could have had a body bomb; he could have had a vest on,” King said.
An intruder jumped the fence at the executive mansion shortly after President Obama departed.
The intruder drove up to the White House gate and pulled into the vehicle screening area.