Insertion quadrupoles with large aperture and high gradient are required to achieve the luminosity upgrade goal of 10{sup 35} cm{sup -2} s{sup -1} at the Large Hadron Collider (LHC). In 2004, the US Department of Energy established the LHC Accelerator Research Program (LARP) to develop a technology base for the upgrade. Nb{sub 3}Sn conductor is required in order to operate at high field and with sufficient temperature margin. We report here on the conceptual design studies of a series of 1 m long 'High-gradient Quadrupoles' (HQ) that will explore the magnet performance limits in terms of peak fields, forces and stresses. The HQ design is expected to provide coil peak fields of more than 15 T, corresponding to gradients above 300 T/m in a 90 mm bore. Conductor requirements, magnetic, mechanical and quench protection issues for candidate HQ designs will be presented and discussed
