Amongst them all, she was the only one with the perquisite to understand the complexity of the situation.

'Perquisite' refers to a benefit or privilege attached to a job, which does not fit the context of understanding complexity. 'Perspicacity' means having keen mental perception and understanding, or discernment, which is the exact quality needed to understand the complexity of a situation. 'Perniciousness' means having a harmful effect. 'Peremptoriness' means insisting on immediate attention or obedience. 'Presumptuousness' means failing to observe the limits of what is permitted or appropriate. Hence, 'perspicacity' is the correct improvement.