Hard drive isn't needed, but then they must offer some kind of storage device that will never need to be changed. Gamecube worked out well for me since their default memory cards are small, so saves are small and thus I was able to buy the interact 8MB memcard and save over 40 games on it.
All games must support up to 1080P, 5.1 dolby digital or dts, and 60 fps. Honestly, I wonder if people would even see a large difference over current consoles if the games were played at 480i?(already so much detail is lost, imagine how much would be on next gen consoles......) Oh, and if a game isn't played in 1080P, it must support anti aliasing. Next consoles will be capable of so many pixels(probably at least 4 billion per second) that they could do 1080P at 60 fps and still have enough power left over to match the current consoles, only problem might be memory amount and bandwidth. Oh and the pixel shader effects used could be a problem, but I do expect realistic lighting out of the next gen consoles.
Free peer to peer Internet play is a must.
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If the console itself is kind of weak from the starting gate, then some sort of sli linkup must be offered to double the power.
All games must support LAN play, that or seperate video outputs to seperate screens.(gamecube could possibly do that, it has two video outputs that can be functional at the same time) How about an external splitter device that can be bought that will split a super hi res image into 4 different 480p images to be output to 4 different tvs.