Clarifying a few things.
The reason why some phones are a dual core is because Qualcomm has the LTE modem integrated into the main chipset with the CPU. This makes it faster, less lag and noise on the communications between the modem and the CPU.
The second is that the dual S4 Krait cores used on these chips are not slower than the quad core chips. Cores are not made equal, those on the S4 chips, called the Krait, are more advanced than the Cortex A9 cores used with quad cores today. The Krait is about 2x faster than the Scorpion core used in previous Qualcomm Snapdragons. Thus each Krait core is nearly as fast as two Cortex A9 cores.
In situations and applications that run one to two threads, the dual core S4 will be faster than the quad core. For a quad core to get its real speed, the application processing has to be broken down to at least four threads so each core can process one of them. Think of a thread like a fuel line to a car cylinder.
In an application that uses multiple threads, like the user interface, there is at least one spare core to handle to the touch and the screen while the other cores can process in the background. In that sense, a quad core might be smoother than a dual core when multiple apps are being multitasked.
Now going back to your question.
The quadcore phones in your selection list are going to be non LTE. You have to debate first whether you really need LTE or not. LTE is definitely faster than 3G, but it does consumer more battery.
If you have chosen a non LTE approach, then your choices boils down to the quad core Galaxy S3 vs. the quad core HTC One X. Between the two, quite franky, its a close match, and it will boil down to your personal taste. Both have different CPUs, the quad core Exynos vs. the quad core Tegra3. But the advantages and disadvantages between the two are not very apparent in real life. More of a benchmarking thing.
If you have taken the LTE approach, then your choices will boil down to the dual core Galaxy S3, HTC One S and One X. All three will have the same CPU, that dual core Qualcomm S4 chip. Its going to boil down with tastes again, if you like Samsungy or the HTCness.
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