The strong magnetic anomalies of Mars require highly magnetic sources in the crust. The bottom of the potentially magnetic layer is constrained by the Curie temperature of its magnetic carriers, and the top of the layer is constrained by the thickness of the uppermost crust that has been demagnetized by the impacts. This paper presents a systematic study of the thermal evolution of the Martian crust and the effects of eight major physical parameters on the thickness of the potentially magnetic layer in the crust. It is shown that the initial upper mantle temperature, the mantle viscosity, and the total radioactive content of Mars are the major parameters that have substantial effects on the thermal state of the crust in the first 1 Gyr of the planet's history. The magnetic source bodies that have been magnetized by the core field during the first 500 Myr are located in the upper about 100--90, 90--80, or 55--45 km of the crust if hematite, magnetite, or pyrrhotite is the major magnetic carrier of the source bodies, respectively. The shock pressures induced in the crust by impacts can demagnetize the uppermost part of the crust. It is demonstrated in this paper that impacts that create craters of diameters larger than ~200 km are capable of demagnetizing the entire crust, and those that create craters of diameters less than ~50 km can demagnetize the upper 10--20 km of the crust. Detailed studies of the secondary thermal remanent magnetization acquired by deeper parts of the crust, in the absence of the core field but in the presence of the magnetic field of the upper crust, suggest that the secondary magnetization has minor effects on the observed magnetic anomalies of Mars.