A one time programmable (OPT) and multiple time programmable (MTP) structure is constructed in a back end of line (BEOL) process using only one, two or three masks. The OTP/MTP structure can be programmed in one of three states, a pre-programmed high resistance state, and a programmable low resistance state and a programmable very high resistance state. In the programmable low resistance state, a barrier layer is broken down during an anti-fuse programming so that the OTP/MTP structure exhibits resistance in the hundred ohm order of magnitude. In the very high resistance state a conductive fuse is blown open during programming so that the OTP/MTP structure exhibits resistance in the mega-ohm order of magnitude. The OTP/MTP structure may include a magnetic tunnel junction (MTJ) structure or a metal-insulator-metal (MIM) capacitor structure.