I'm not sure about that but I do not think we will need to download the entire network of blockchain when we use the lightning network.
If you want to open a channel on the lightning network you need to have the full blockchain stored.
But there will be lightweight clients to access the LN, for sure. Just like it was with wallets.
To have a synced wallet on your PC you need the blockchain. Or you rely on a 3rd party to give you the information you need.
The same applies to the LN. To open up a channel without the need of trusting anyone, you need the whole blockchain.
But if you rely on information from 3rd party server you can use a lightweight client (e.g.
https://github.com/mit-dci/lit).
Note, that this client is still being developed and every software on the LN should be used carefully.