There are numerous references to improved foraging, deposit and storage of honey with multiple entrances.
There is a good article from Rusty (HoneyBeeSuite) on the subject of additional entrances.
There is some further reading here by Michael Bush.
Tautz (1996) reported that the nature of the floor on which the bees dance has considerable influence on the recruitment of nestmates to a food source. Dancers on combs with open empty cells (supers) recruit three times as many nestmates to a food source as dancers on capped BROOD cells.
Contact rates have been shown to regulate the allocation of workers among tasks in colonies of harvester ants (Pinter-Wollmanetal. 2013), and honeybees (Seeley1989). An increased number of workers in a colony, which may increase the worker contact rate, may also be a stimulus for swarming, the female mode of reproduction in honeybees (reviewed by Grozinger et al. 2013). But it remains unknown whether contact rates are used to assess the number of workers in a colony, and it is entirely possible that workers regulate their contact rates (Gordon et al. 1993) rather than being regulated by them.
Guarding Behaviour and Strengthening of Colony Odour
Examination by guards at the honey bee entrance of definite intruders may last for a few seconds only and may then be immediately followed by mauling; on the other hand, very often a number of guard bees will each, singly or together, examine an intruder for half-a-minute or more and leave her without attempting to maul her. It appears that in the latter cases the guards are uncertain of the identity of the intruders.
This conclusion regarding the importance of scent in the recognition by bees of members of their own and other colonies is strengthened by the
observations of KALMUS and RIBBANDS (1952) who showed in experiments with foraging bees that the bees of every colony have their own distinctive odour and that foraging bees recognise and prefer the scent produced by members of their own colony.
Elishia Gallup, The American Bee Journal 1867, Volume 3, Number 8 pg 153
"Again if you see a box hive with a crack in it from top to bottom large enough to put your fingers in, the bees are all right in nine cases out of ten. The conclusion I have come to is this, that with upward ventilation without any current of air from the bottom of the hive, your bees will winter well without any cobs."
Reasons for top entrances supplementing smaller bottom entrances
You can keep bees fine without higher entrances, but they do eliminate the following problems:
mice and other rodents;
cluster to far removed from entrance;
dead bees blocking the exit in winter;
condensation on the lid in winter, snow ;
blocking the exit in winter;
grass blocking the exit the rest of the year;
It also allows you to buy inexpensive floors as well as decrease hive costs.
THE MOST IMPORTANT REASON FOR A HIGHER ENTRANCE IS THAT IT IS CLOSER TO THE WINTER CLUSTER, THEREBY:
ALLOWING FOR EARLIER CLEANSING FLIGHTS;
IDENTIFYING COLONIES THAT MAY HAVE DEVELOPED A PROBLEM OVER THE WINTER;
IMPROVING COLONY HEALTH IN THE SPRING BY ALLOWING FRESH NUTRIENTS TO BE DEPOSITED CLOSE TO THE DEVELOPING BROOD NEST
A Revolution in Bee Hive Technology and Management