Q:

Is "Sequencing Breakthrough" applied only to a single player

The Sequencing Breakthrough event indicates to place the card face up in front of "this player". The text, however, indicates the effect is applied on the "next cure".
All of the other long-term events (e.g. Commercial Travel Ban) are somehow tied to the player during whose turn the event was played. Is Sequencing Breakthrough similar - does it apply only to the player it was played by?
I've been playing it as a bonus on whichever cure is discovered next, regardless of who discovers it.

A:

You have been playing correctly. The effect applies to the next 'Discover A Cure' action, regardless of which player performs it.
The card is placed in front of the player for two reasons:

it remains in effect (the obvious and helpful indicator so that players can easily check whether it applies)
it is kept out of the discard pile (to prevent other abilities and card effects from being used to create a "stack" of Sequencing Breakthroughs)

