Turns over but doesn't start would rule out battery and starter but leaves to many other possible causes to guess at with out being able to see the car first hand. The car needs mechanical integrity/ develop compression to start, so the timing belt needs to in tact to keep the valves in synchronization with the crankshaft so they open and close at the appropriate time.
Next the engine needs ignition to set fire to the fuel. The ignition distributor on the Corolla is driven by the camshaft so if the rotor in the distributor spins as the engine is cranked over at least you know the timing belt is still attached. Remove a spark plug wire put an old spark plug in the plug wire end, hold the metal case of the spark plug (that screws into the cylinder head) firmly against any metal surface of the engine for ground/earth. Get a friend to crank over the engine. Do you see a spark jump across the spark plug? If not the ignition system may have a failed ignition coil, ignition rotor, distributor cap or rotor, or loss of electrical power to the ignition or failed sensors in the distributor.
Electrical sensors in the distributor need to tell the fuel injection computer that the engine is cranking or running to operate the fuel pump. With the engine cranking fuel pressure should be prescient in the fuel rail in the back of the engine. If you have fuel pressure it say the sensors in the distributor are actually working.
Unfortunately to many things could cause your no start condition.
I am working on a long detailed article with pictures to address this question as it is frequently asked.
I had a bad fuel pump and the car would turn and after while of turning the car would start. After I changed the fuel pump everything was working again. however, after a while it started doing the same thing. Im thinking something electrical. What can it be??