Before revision:  Namely, we demonstrate that any T-query algorithm using S qubits of memory must satisfy a tradeoff of T^3 S \geq \Omega( N^4)for finding \Theta(N) collisions in a random function. We conjecture that this result can be improved to ] T^2 S \geq \Omega(N^3), and we show that it would imply a T^2 S \geq (N^2) tradeoff for Element Distinctness ]  .
After revision:  Namely, we demonstrate that any T-query algorithm using S qubits of memory must satisfy a tradeoff of T^3 S \geq \Omega( K^3 N). Classically, the same question has only been settled recently by Dinur ] T^2 S \geq \Omega(N^3), and we show that it would imply a T^2 S \geq (N^2) tradeoff for Element Distinctness ] .