MTG: Abilities lost when exiled?
Your creatures will not have hexproof, but they will be unblockable
Glaring Spotlight's activated ability creates 2 continuous effects. One grants hexproof, so it changes the objects' &quot;abilities&quot; characteristic. The other  (&quot;creatures you control can't be blocked&quot;) modifies the rules of the game. Because of that difference, the first is locked in to the set of objects and will not affect other objects (such as those objects that are created after the blink from Legion's Initiative), whereas the second affects all applicable objects as long as it's active.

109.3. An object’s characteristics are name, mana cost, color, color indicator, card type, subtype, supertype, rules text, abilities, power, toughness, loyalty, hand modifier, and life modifier.
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.
611.1. A continuous effect modifies characteristics of objects, modifies control of objects, or affects players or the rules of the game, for a fixed or indefinite period.
611.2. A continuous effect may be generated by the resolution of a spell or ability.
611.2c If a continuous effect generated by the resolution of a spell or ability modifies the characteristics or changes the controller of any objects, the set of objects it affects is determined when that continuous effect begins. After that point, the set won’t change. (Note that this works differently than a continuous effect from a static ability.) A continuous effect generated by the resolution of a spell or ability that doesn’t modify the characteristics or change the controller of any objects modifies the rules of the game, so it can affect objects that weren’t affected when that continuous effect began. If a single continuous effect has parts that modify the characteristics or changes the controller of any objects and other parts that don’t, the set of objects each part applies to is determined independently.