Get count of new entries since last visit
You can use the cookies Plugin (or for Craft 2)  to store the value of the timestamp when the users has visited the page for the last time. Or you can create your custom code like explained in the question: How to set a cookie

You can get the current timestamp like it's stated in the docs 

{% set currentTimestamp = now|date('U') %}


And store the cookie like explained

{# Set the cookie using 'set' variable #}
{% do craft.cookies.set( NAME, VALUE, DURATION, PATH, DOMAIN, SECURE, HTTPONLY) %}
{% set yourTimeStamp = getCookie( NAME ) %}


Then you'll search for entries after the specific date

{% set newEntries = craft.entries.postDate('&gt;= ' ~ yourTimeStamp)|length %}