Can you activate a planeswalker's loyalty ability the same turn it enters the battlefield?

Planeswalkers can activate their loyalty abilities on the same turn they enter the battlefield. The 'summoning sickness' rule (CR 302.6) only restricts creatures from attacking or using activated abilities that include the tap or untap symbol. Planeswalkers are not creatures (unless a card specifically makes them one), so that restriction does not apply to them.

Loyalty abilities are a special kind of activated ability identified by a loyalty cost (e.g., [+1], [−2]). CR 606.3 states that a player may activate a loyalty ability of a planeswalker they control any time they could activate a sorcery-speed ability, as long as no other loyalty ability of that permanent has been activated that turn.

The key rule is CR 306.5b: a loyalty ability can be activated only at sorcery speed and only once per turn per planeswalker. There is no additional restriction preventing activation in the same turn the planeswalker entered the battlefield.

Example: You cast Liliana of the Veil on your main phase. It resolves and enters with 3 loyalty counters. You may immediately activate her [+1] ability that same turn, bumping her to 4 loyalty and forcing each player to discard a card.

One condition to be aware of: if a planeswalker enters the battlefield under your control during an opponent's turn (e.g., via a flash effect or a triggered ability), you cannot activate its loyalty ability until your next turn, because loyalty abilities are sorcery-speed (CR 606.3a).