All travelers need a passport that's valid for at least six months from the date of entry. You're also required to provide evidence of onward travel and prove that you have enough money to pay for your trip. These last two requirements are rarely checked, but you should have proof anyway just in case you're stopped.
Most travelers can stay in Ecuador for 90 days without a visa. If you want to stay for longer than 90 days, you need to get a visa before your trip. You can do so at an Ecuadorian consulate.
While you're in Ecuador, you're technically required to carry your passport at all times. However, having a photocopy of your passport will usually suffice and is much safer than carrying your real passport around with you all day. If possible, store your real passport in a safety deposit box at your hotel.
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