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Why can you not match orders in parallel using logarithmic reduction, the same way you would sort in parallel? Is it that there is not enough other computation being done other than sorting by time and price?

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It's an inherently serial problem and regulations require it to be that way. Users who submit first want their orders to be the one that crosses.

Stupid question, since I know next to nothing about exchanges and regulations... but couldn't you just process serially by security?

E.G. if user A wants to buy Apple stock and user B wants to buy Facebook stock, does it matter which order came first? And if yes, why?

I think that's allowed but this is where my meagre expertise runs out. You normally have to process orders serially or at least using algorithms that yield the exact same outcome that serial execution would give, but only within a single order book.

I think it is the temporal aspect of order matching - for exchanges it is an inherently serial process.

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